ON CALCULATIONS OF ZEROS OF VARIOUS L-FUNCTIONS

By Hiroyuki Yoshida

Introduction. As we have shown several years ago [Y2], zeros of L(s, Δ) and L^{(2)}(s, Δ) can be calculated quite efficiently by a certain experimental method. Here Δ denotes the cusp form of weight 12 with respect to SL(2, Z) and L(s, Δ) (resp. L^{(2)}(s, Δ)) denotes the standard (resp. symmetric square) L-function attached to Δ. The purpose of this paper is to show that this method can be applied to a wide class of L-functions so that we can obtain precise numerical values of their zeros.^{1}

We organize this paper as follows. In §1, we shall describe basic features of our method of calculation, which is repeated applications of partial summation. In §2, we shall study the r-th symmetric power L-function L^{(r)}(s, Δ) attached to Δ. Since the cases r = 1, 2 are discussed in [Y2], we shall exclusively treat the cases r = 3, 4. In §3, we shall study the L-functions attached to modular forms of half integral weight. These L-functions do not have Euler products. Naturally the Riemann hypothesis fails for them; we shall find many zeros off the critical line, though major part of zeros lie on the critical line. We shall also calculate the location of these zeros off the critical line. Though there is some hope to find relations among zeros of L-functions of two modular forms which are in the Shimura correspondence, no explicit results came out so far.

In §4, we shall study L-functions attached to Hecke characters of non-A_{0} type of real quadratic fields. D.A. Hejhal showed great interest to make experiments in this case, since coefficients are non-computable combinatorially; hence there is a slight possibility that the Riemann hypothesis may break down for these Lfunctions. We have made experiments on 44 cases summarized in Table 4.3; so far no counterexamples are found.

In §5, we shall study the Artin L-function attached to a 4-dimensional nonmonomial representation of Gal (\overline{Q}/Q). In §6, we shall discuss the controle of error estimates in our calculation. In §7, we shall consider the explicit formula for the L-function attached to a modular form of weight 8 with respect to Γ_{0}(2). We shall compare both sides of the explicit formula numerically. In §8, we shall present sample programs to compute values of L-functions, which may be convenient for the reader. In §9, we shall formulate a conjecture which emerged during the process of our experiments.

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∗ During the final stage of writing this paper, the author was at MSRI supported in part by NSF grant ♯DMS9022140.

1After the publication of [Y2], H. Ishii [Is] published a table of zeros of standard L-functions attached to modular forms for 15 cases. It also comes to the author’s notice that a program of the calculation of zeros of L(s, Δ) is included in “Mathematica” package, following the method of [Y2].

Most of sections have attached tables to show results explicitly. Concerning actual computations, we have used “UBASIC ” created by Y. Kida. (It was not available when we wrote [Y2].) The calculation was done by personal computers which are not necessarily so fast. However our experiments extended over long time (about three years) and UBASIC is quite fast (compared with some other softwares) for numerical calculations, the author thinks that our tables are fairly extensive.

A motivation in these calculations has been to find non-trivial “functorial” properties which may exist among zeros of L-functions, as was hinted in [Y1]. Though our experiments are not successful in this regard, conjectures stemmed from them are formulated in §9.

We can pursue these calculations still further. The topics which may be included in this paper are:

1) The Hasse-Weil zeta functions of algebraic curves, for example y^{2} = x^{5} − x + 1.

2) The Dirichlet series ∑_{n=1}^{∞} \frac{nα − [nα] − 1/2}{n^{s}} studied by Hecke [H], where α is a real

irrational number.

3) Applications to Riemann-Siegel type formulas.

4) Calculations of critical values of L-functions. Our results on these topics are still fragmentary, so the full discussion should be postponed to future occasions.

Notation. For a complex number z, we denote by ℜ(z) (resp. ℑ(z)) the real (resp. imaginary) part of z. The letter q stands for exp(2π\sqrt{−1}z) when it is clear from the context. For modular forms, we follow the notation in Shimura [Sh1].

§1. An overview on our method of calculations

Let

(1.1)  L(s) = ∑_{n=1}^{∞} a_{n}n^{−s}

be a Dirichlet series which is absolutely convergent when ℜ(s) > σ for some σ > 1. In this paper, we shall consider only such L(s) which can be analytically continued to the whole complex plane as an entire function and satisfies a functional equation of the form

(1.2)  R(k − s̄) = κ\overline{R(s)}.

Here κ is a constant of absolute value 1, k > 0,

R(s) = N^{s} ∏_{i=1}^{m} Γ(b_{i}s + c_{i})L(s)

with N > 0, b_{i} > 0, c_{i} ∈ C. We note that (1.2) is equivalent to

(1.3)  R(k − s) = κR(s)

if a_{n} ∈ R, c_{i} ∈ R for all n and i. Put κ^{−1} = κ_{1}^{2} with some κ_{1} ∈ C. By (1.2), we have

(1.4)  κ_{1}R(s) ∈ R  if ℜ(s) = k/2.

Take any δ > 0. For T > 0, let N(δ, T) denote the number of zeros of R(s) counted with multiplicity in the domain

−δ ≤ ℜ(s) ≤ k + δ, 0 ≤ ℑ(s) ≤ T.

Let D be the rectangle whose vertices are −δ, k + δ, k + δ + iT, −δ + iT and let C denote the contour ∂D taken in positive direction. By the argument principle, we have

(1.5)  N(δ, T) = \frac{1}{2πi} ∫_{C} \frac{R′(s)}{R(s)} ds,

assuming that neither zeros nor poles of R(s) lie on C. Let C_{1} denote the portion of C from k/2 to k/2 + iT. By the functional equation (1.2), we obtain

(1.6) N(δ, T) = π^{−1}Δarg R(s) = π^{−1}Δ(arg N^{s} ∏_{i=1}^{m} Γ(b_{i}s + c_{i})) + π^{−1}Δ(arg L(s)),

where Δarg denotes the variation of the argument on C_{1}, i.e., from s = k/2 to k/2 + iT along k/2 to k + δ, k + δ to k + δ + iT, k + δ + iT to k/2 + iT. Set^{2}

ϑ(T) = Δarg (N^{s} ∏_{i=1}^{m} Γ(b_{i}s + c_{i})).

Assume

ℜ(c_{i}) > −b_{i}k/2 for 1 ≤ i ≤ m.

Then since b_{i} > 0, N^{s} ∏_{i=1}^{m} Γ(b_{i}s + c_{i}) has neither zeros nor poles in the domain ℜ(s) ≥ k/2. Hence ϑ(T) is equal to the variation of the argument of N^{s} ∏_{i=1}^{m} Γ(b_{i}s+ c_{i})) on the line segment [k/2, k/2 + iT]. We note that ϑ(T) can be computed in high precision very easily using Stirling’s formula (cf. [WW], p. 252) combined with the relation Γ(s + 1) = sΓ(s). We obtain

(1.7)  N(δ, T) = π^{−1}ϑ(T) + π^{−1}Δ(arg L(s)).

Now let us consider the case when R(s) has zeros in (−δ, k + δ). Let r denote the number of zeros of R(s) i.e., of L(s), in this interval counted with multiplicity. Then (1.7) holds with the modification

(1.8)  N(δ, T) − \frac{r}{2} = π^{−1}ϑ(T) + π^{−1}Δ(arg L(s)).

2When it is clear from the context, we shall use ϑ(T) for the “phase factor” of this type in the following sections without further explanation.

Here Δ(arg L(s)) is counted by dividing C_{1} into a finite number of paths removing real zeros of L(s) and summing the variations of the argument of L(s) on each of them. The validity of (1.8) can be seen by modifying C by small semi-circles which detour the real zeros of L(s).

Throughout the paper, to compute L(s), we shall employ our method given in [Y2], which is repeated applications of Abel’s partial summation. Set

s_{n}^{(0)} = a_{n}, u_{n}^{(0)} = n^{−s}

and define s_{n}^{(l)}, u_{n}^{(l)} recursively by

(1.9)  s_{n}^{(l)} = ∑_{m=1}^{n} s_{m}^{(l−1)},  u_{n}^{(l)} = u_{n}^{(l−1)} − u_{n+1}^{(l−1)},  l ≥ 1.

Put S_{N}^{(l)} = ∑_{n=1}^{N} s_{n}^{(l)}u_{n}^{(l)}. Then we have

(1.10)  S_{N}^{(l)} = S_{N}^{(l−1)} − s_{N}^{(l)}u_{N+1}^{(l−1)}.

As we have seen in [Y2], in several cases, S_{N}^{(l)} seems to approximate L(s) amazingly well when we choose N and l sufficienly large. In the succeeding sections, we shall present various types of L-functions which can be treated in more or less similar fashion. The efficacy of our method seems to depend strongly on the arithmetical nature of the coefficients a_{n} of a Dirichlet series L(s).

We shall conclude this section by technical remarks concerning actual computations of S_{N}^{(l)}. As the first step, we should construct a table of a_{n}. For Dirichlet series considered in this paper, this step can be achieved rather easily. Since we

can compute S_{N}^{(l)} from S_{N}^{(0)} by (1.10), the computation of S_{N}^{(0)} = ∑_{n=1}^{N} a_{n}n^{−s} is the

substantial and the most time consuming part of our calculation. However usually s_{n}^{(l)} becomes very large and u_{n}^{(l)} very small when l increases. Therefore it is indispensable to perform the actual computation in high precision. For u_{n}^{(l)}, the following formula (1.11) should preferably be used than to compute it directly from the definition.

(1.11)  \begin{aligned}u_{N}^{(l)} & = N^{−s} ∑_{k=l}^{∞}( ∑_{m=1}^{l}(−1)^{m}\binom{l}{m}m^{k}) \\ & (−1)^{k} \frac{s(s + 1) ⋯ (s + k − 1)}{k!}N^{−k}, \end{aligned}  N > l ≥ 1.

If we replace ∑_{k=l}^{∞} by ∑_{k=l}^{L}, the error is less than 2^{l} \frac{|s(s+1)···(s+L)|}{(L+1)!}(\frac{l}{N})^{L+1}|N^{−s}| if ℜ(s) ≥ −L − 1.

§2. L-functions attached to symmetric tensor representations of GL(2)

Let f(z) = ∑_{n=1}^{∞} c_{n}e^{2πinz} ∈ S_{k}(SL(2, Z)) be a normalized common eigenfunction of Hecke operators. The L-function L(s, f) = ∑_{n=1}^{∞} c_{n}n^{−s} attached to f converges absolutely when ℜ(s) > \frac{k+1}{2} and has the Euler product

L(s, f) = ∏_{p}(1 − c_{p}p^{−s} + p^{k−1−2s})^{−1}.

Put

1 − c_{p}X + p^{k−1}X^{2} = (1 − α_{p}X)(1 − β_{p}X)

with α_{p}, β_{p} ∈ C, where X is an indeterminate. For a positive integer r, we define an Euler product

(2.1)  L^{(r)}(s, f) = ∏_{p}[(1 − α_{p}^{r}p^{−s})(1 − α_{p}^{r−1}β_{p}p^{−s}) ⋯ (1 − β_{p}^{r}p^{−s})]^{−1}

which converges absolutely when ℜ(s) > \frac{r(k−1)}{2} +1. It is conjectured that L^{(r)}(s, f) can be analytically continued to the whole complex plane as an entire function and satisfies a functional equation. The conjectural functional equation of L^{(r)}(s, f) takes the following form (cf. Serre [Se]). If r is odd, put r = 2m − 1,

(2.2)  R^{(r)}(s, f) = (2π)^{−ms} ∏_{i=0}^{m−1} Γ(s − i(k − 1))L^{(r)}(s, f),

(2.3)  ϵ_{r} = (\sqrt{−1})^{m+(k−1)m^{2}}.

If r is even, put r = 2m,

(2.4)  \begin{aligned}R^{(r)}(s, f) = π^{−s/2}(2π)^{−ms} & ( ∏_{i=0}^{m−1} Γ(s − i(k − 1))) \\ & Γ(\frac{s − m(k − 1) + δ}{2} )L^{(r)}(s, f), \end{aligned}

(2.5)  ϵ_{r} = (\sqrt{−1})^{m+(k−1)m(m+1)+δ},

where δ = 0 (resp. 1) if m is even (resp. odd). Then the functional equation

(2.6)  R^{(r)}(s, f) = ϵ_{r}R^{(r)}(r(k − 1) + 1 − s, f)

is predicted. A quick way to see (2.6) is as follows. Let M_{f} be the motive of rank 2 over Q attached to f. We see that the Hodge realization of M_{f} corresponds to the two dimensional representation

ρ = Ind(ψ_{k}; W_{C} ⟶ W_{R,C})

of W_{R,C}. Here W_{C} = C^{×}, W_{R,C} is the Weil group of C over R and ψ_{k} is the quasi-character ψ_{k}(x) = x^{−(k−1)} of W_{C}. Let σ_{r} : GL(2) ⟶ GL(r + 1) be the symmetric tensor representation of degree r and put ρ_{r} = σ_{r} ∘ ρ. Then we find

(2.7) ρ_{r} ≅ ⊕_{i=0}^{m−1}Ind(x ⟶ x^{−(r−i)(k−1)}x̄^{−i(k−1)}; W_{C} ⟶ W_{R,C}), r = 2m − 1,

(2.8)

ρ_{r} ≅ ⊕_{i=0}^{m−1} Ind(x ⟶ x^{−(r−i)(k−1)}x̄^{−i(k−1)}; W_{C} ⟶ W_{R,C})
⊕ {(x ⟶ |x|^{−m(k−1)}(sgn x)^{m(k−1)}) ∘ t}, r = 2m,

where t denotes the transfer map from W_{R,C} to R^{×}. The gamma factor and the constant ϵ_{r} of the functional equation can be calculated as the usual gamma factor and the constant attached to the representation ρ_{r} of W_{R,C}; hence we obtain (2.2) ∼ (2.6).

We refer the reader to Shahidi [Sha1], [Sha2] for what are known on these symmetric power L-functions, in more general cases.

Let Δ(z) = q ∏_{n=1}^{∞}(1 − q^{n})^{24} ∈ S_{12}(SL(2, Z)), q = e^{2π\sqrt{−1}z}. The calculation of zeros of L^{(r)}(s, Δ) for r = 1, 2 is given in [Y2]. We consider the case r ≥ 3. To compute L^{(r)}(s, Δ), we modify our summation method slightly in the following way. Fix r, choose v = v_{r} > 0 and set

L^{(r)}(s, Δ) = ∑_{n=1}^{∞} a_{n}n^{−s} = ∑_{n=1}^{∞}(a_{n}n^{−v})n^{−(s−v)}.

Put

s_{n}^{(0)} = a_{n}n^{−v}, u_{n}^{(0)} = n^{−(s−v)}

and define s_{n}^{(l)}, u_{n}^{(l)} recursively by (1.9). We set S_{N}^{(l)} = ∑_{n=1}^{N} s_{n}^{(l)}u_{n}^{(l)}. It turns out that a suitable choice of v depending on r yields good results. We can interpret this as the neutralization of the effect of extremely large value of s_{n}^{(l)} and extremely small value of u_{n}^{(l)}.

As the first example, let r = 3. We take v = 8. For s = 17 + it, t = 20, the values of

R_{j} = ℜ(exp(iϑ(t))S_{N}^{(j)}), I_{j} = ℑ(exp(iϑ(t))S_{N}^{(j)})

are given in Table 2.1. In Table 2.2, we give the values of t_{n} the n-th zero of L^{(3)}(s, Δ), s = 17 + it on the critical line for 0 ≤ t ≤ 40.

Next we apply our summation method to L^{(4)}(s, Δ) taking v = 12. For s = \frac{45}{2} + it, t = 10, the values of

R_{j} = ℜ(exp(iϑ(t))S_{N}^{(j)}), I_{j} = ℑ(exp(iϑ(t))S_{N}^{(j)})

are given in Table 2.3. In Table 2.4, we give the values of u_{n} the n-th zero of L^{(4)}(s, Δ), s = \frac{45}{2} + iu on the critical line for 0 ≤ u ≤ 20.

We can see, by the same technique as will be given in §3 and §4, that the Riemann hypothesis holds for L^{(3)}(s, Δ) (resp. L^{(4)}(s, Δ)) in the range 0 ≤ ℑ(s) ≤ 40 (resp. 0 ≤ ℑ(s) ≤ 20 ) and that the zeros 17 + it_{n} (resp. \frac{45}{2} + iu_{n}) are simple.

§3. Modular forms of half integral weight

Put

θ(z) = ∑_{n∈Z} e(n^{2}z) = 1 + 2 ∑_{n=1}^{∞} q^{n^{2}}, η(z) = e(z/24) ∏_{n=1}^{∞}(1 − e(nz)).

By Shimura, [Sh2], (4.1), we have

dim S_{8}(Γ_{0}(2)) = 1, dim S_{9/2}(Γ_{0}(4)) = 1

and (η(z)η(2z))^{8} (resp. θ(z)^{−3}η(2z)^{12}) spans S_{8}(Γ_{0}(2)) (resp. S_{9/2}(Γ_{0}(4))). Put

f(z) = (η(z)η(2z))^{8} = ∑_{n=1}^{∞} a_{n}q^{n}, g(z) = θ(z)^{−3}η(2z)^{12} = ∑_{n=1}^{∞} c_{n}q^{n},

L(s, f) = ∑_{n=1}^{∞} a_{n}n^{−s}, L(s, g) = ∑_{n=1}^{∞} c_{n}n^{−s},

R(s, f) = 2^{s/2}(2π)^{−s}Γ(s)L(s, f), R(s, g) = 2^{s}(2π)^{−s}Γ(s)L(s, g).

Then f and g are in the Shimura correspondence; L(s, f) and L(s, g) can be analytically continued to the whole complex plane as entire functions and satisfy the functional equations

(3.1)  R(s, f) = R(8 − s, f),  R(s, g) = R(9/2 − s, g).

This example is described in detail in [Sh2], §4. For t > 0, let ϑ_{f}(t) (resp. ϑ_{g}(t)) denote the variation of the argument of 2^{s/2}(2π)^{−s}Γ(s) (resp. 2^{s}(2π)^{−s}Γ(s)) from 4 to 4 + it (resp. 9/4 to 9/4 + it).

For L(s, f), s = 4 + it, t = 100, the values of

R_{j} = ℜ(exp(iϑ_{f}(t))S_{N}^{(j)}), I_{j} = ℑ(exp(iϑ_{f}(t))S_{N}^{(j)})

are given in Table 3.1.

For L(s, g), s = \frac{9}{4} + it, t = 100, the values of

R_{j} = ℜ(exp(iϑ_{g}(t))S_{N}^{(j)}), I_{j} = ℑ(exp(iϑ_{g}(t))S_{N}^{(j)})

are given in Table 3.2.

By our method, we can compute zeros of L(s, f) and of L(s, g) on the critical line with sufficient accuracy observing sign changes of e^{iϑ_{f} (t)}L(4+it, f) and e^{iϑ_{g}(t)}L(\frac{9}{4} + it, g). In Table 3.3, we list the n-th zero t_{n} of L(s, f), s = 4 + it in the range 0 ≤ t ≤ 100. In Table 3.4, we list the n-th zero u_{n} of L(s, g), s = \frac{9}{4} + iu in the range 0 ≤ u ≤ 100.

Now let us examine the Riemann hypothesis for L(s, f). We see f(iy) > 0 for y > 0 by the product expansion of the η-function. By the integral representation

(2π)^{−s}Γ(s)L(s, f) = ∫_{0}^{∞} f(iy)y^{s−1} dy,

we see that L(s, f) > 0 for s > 0. For T > 0, let N(T) denote the number of zeros of L(s, f) counted with multiplicity in the domain

|ℜ(s) − 4| < 1/2, 0 ≤ ℑ(s) ≤ T.

By (1.7) taking δ = 1/2, we have

(3.2)  N(T) = π^{−1}ϑ_{f}(T) + π^{−1}Δarg(L(s, f)).

Since L(s, f) ≠ 0 if ℜ(s) − 4 ≥ 1/2, Δarg(L(s, f)) equals the variation of the argument of L(s, f) along the line segments L_{1} = [4, 4 + \frac{1}{2} + μ], L_{2} = [4 + \frac{1}{2} + μ, 4 + \frac{1}{2} + μ + iT], L_{3} = [4 + \frac{1}{2} + μ + iT, 4 + iT] for any μ > 0. Take T = 100, μ = 1. Then we have π^{−1}ϑ_{f}(100) = 69.0171 · · ·. Hence if we can show |Δarg(L(s, f))| < π/2, we can conclude N(T) = 69. For this purpose, it suffices to show that ℜ(L(s, f)) > 0 when s ∈ L_{i}, i = 1, 2, 3. For L_{1}, we have shown this fact above. For L_{2}, this fact can be proved as in [Y2], §4. For L_{3}, we divide it into 150 small intervals and appeal to our heuristic calculation. We have observed

ℜ(L(s, f)) > 0.83 on L_{3}.

Thus we conclude that the Riemann hypothesis holds for L(s, f) when 0 ≤ ℑ(s) ≤ 100. All the zeros are simple.

Now let us consider zeros of L(s, g). We have g(iy) > 0 for y > 0 since θ(iy) > 0, η(iy) > 0 for y > 0. By the integral representation

(2π)^{−s}Γ(s)L(s, g) = ∫_{0}^{∞} g(iy)y^{s−1} dy,

we see that L(s, g) > 0 for s > 0. For T > 0, let N(T) denote the number of zeros of L(s, g) counted with multiplicity in the domain^{3}

|ℜ(s) − 9/4| ≤ 2, 0 ≤ ℑ(s) ≤ T.

By (1.7) taking δ = 2, we have

N(T) = π^{−1}ϑ_{g}(T) + π^{−1}Δarg(L(s, g)).

Here Δarg(L(s, g)) denotes the variation of the argument of L(s, g) along the line segments L_{1} = [9/4, 17/4], L_{2} = [17/4, 17/4 + iT], L_{3} = [17/4 + iT, 9/4 + iT]. Take T = 100. Then we have π^{−1}ϑ_{g}(100) = 79.1885 · · ·. Dividing L_{2} and L_{3} into small intervals, we have observed π^{−1}Δ(argL(s, g)) = 0.8114 · · ·. Thus we obtain N(T) = 80. On the otherhand, we have obtained only 54 zeros on the critical line. Therefore, assuming that these zeros are simple, there must exist 13 zeros in the right-hand side of the critical line: 9/4 < ℜ(s) ≤ 17/4, 0 ≤ ℑ(s) ≤ 100. These zeros, together with those in 100 ≤ ℑ(s) ≤ 150, are given in Table 3.5.

3The zero free region of L(s, g) is non-trivial. Here we content ourselves by regarding |ℜ(s) − 9/4| ≤ 2 is “sufficiently wide”.

Our method of calculation of these exceptional zeros is as follows. Let us consider a box B given by d_{1} ≤ ℜ(s) ≤ d_{2}, h_{1} ≤ ℑ(s) ≤ h_{2}. By the argument principle, we can determine whether L(s, g) has a zero incide B or not. First we find a box B in which L(s, g) has zeros by trial and error. Then dividing B into sub-boxes and applying the principle above successively, we can obtain a good approximation for a zero inside of B.^{4}

The Riemann hypothesis does not hold for L(s, g). This should be of no surprise since L(s, g) does not have an Euler product.

We shall study one more example of modular forms of half integral weight. For k ≥ 1, put

G_{k}(z) = \frac{1}{2}ζ(1 − k) + ∑_{n=1}^{∞} σ_{k−1}(n)q^{n}, σ_{k−1}(n) = ∑_{d|n,d>0} d^{k−1}.

Let

δ(z) = \frac{60}{2πi}(2G_{4}(4z)θ′(z) − G′_{4}(4z)θ(z)) = ∑_{n=1,n≡0,1 mod 4}^{∞} c(n)q^{n}.

Then we have δ ∈ S_{13/2}(Γ_{0}(4)) (cf. Kohnen-Zagier [KZ], p. 177) and δ corresponds to Δ(z) = η(z)^{24} ∈ S_{12}(SL(2, Z)) under the Shimura correspondence. The values of c(n) can easily be computed by

c(n) = ω(\sqrt{n}) · n + 120 ∑_{m=1}^{[ \frac{n−1}{4} ]} ω(\sqrt{n − 4m})σ_{3}(m)(2n − 9m) − 15nσ_{3}(n/4),

where

ω(x) = \{ \begin{matrix}1  if x ∈ Z, \\ 0  if x ∉ Z. \end{matrix}

Let

δ_{0}(z) = ∑_{n≡0 mod 4} c(n)q^{n/4}.

Then we have δ_{0} ∈ S_{13/2}(Γ_{0}(4)) and

δ(\frac{i}{4y}) = −\sqrt{2}y^{6+1/2}δ_{0}(iy), y > 0.

(cf. [KZ], p. 190.) Put

h_{±}(z) = (δ ∓ 2^{−6}δ_{0})(z), R(s, h_{±}) = 2^{s}(2π)^{−s}Γ(s)L(s, h_{±}).

Then we have

h_{±}(\frac{i}{4y}) = ±(2y)^{6+1/2}h_{±}(iy), y > 0.

4It is more efficient to use a variant of the Newton method once we get a rough approximation.

After finding a precise location, final check should be done by the method described above.

Hence we obtain the functional equations for the entire functions R(s, h_{±}):

R(\frac{13}{2} − s, h_{±}) = ±R(s, h_{±}).

We have computed zeros of L(s, h_{±}) on the critical line ℜ(s) = 13/4 in the range 0 ≤ ℑ(s) ≤ 100. The results are given in Table 3.6. There u_{n}^{±} denotes the n-th zero of L(s, h_{±}) for s = \frac{13}{4} + iu.

In Table 3.7, we also list zeros of L(s, h_{±}) not on the critical line which are searched in the region 13/4 < ℜ(s) ≤ 25/4, 0 ≤ ℑ(s) ≤ 100; ρ_{i}^{±} denotes a zero of L(s, h_{±}). It is a very interesting phenomenon that L(s, h_{−}) has much more zeros outside of the critical line compared with L(s, h_{+}).

§4. L-functions attached to Hecke characters of infinite order of real quadratic fields

In this and the next section, we shall study two types of L-functions which are closely related to algebraic number fields. We can still apply our method of calculation described in §1 efficiently. However the situation changes drastically. The repeated application of partial summation does not yield good results beyond rather limited number of times. Thus our calculation cannot be as accurate as in the case of modular forms treated in §2 and §3.

Let k be a real quadratic field. For simplicity, we assume that the class number of k is 1. Let D be the discriminant and ϵ be the fundamental unit of k. Let k_{A}^{×} denotes the idele group of k. For a finite place v of k, let k_{v} denote the completion of k at v and 𝔒_{v} denote the ring of integers of k_{v}. Since k is of class number 1, we have

(4.1)  k_{A}^{×} = k^{×}(∏_{v} 𝔒_{v}^{×} × R^{×} × R^{×})

where v extends over all finite places of k. Let χ = ∏_{v} χ_{v} be an unramified unitary Hecke character of k_{A}^{×}. Let σ_{1} (resp. σ_{2}) be the identical (resp. non-identical) isomorphism of k into R and let ∞_{1} (resp. ∞_{2}) be the corresponding archimedean place of k. As unitary characters of R^{×}, χ_{∞_{1}} and χ_{∞_{2}} take the following form:

(4.2)  χ_{∞_{j}}(x) = sgn(x)^{m_{j}}|x|^{iv_{j}} for x ∈ K_{∞_{j}}^{×} ≅ R^{×}, j = 1, 2,

where m_{j} = 0 or 1, v_{j} ∈ R. By (4.1), we see that χ is completely determined by χ_{∞_{1}} and χ_{∞_{2}}. Since χ is trivial on k^{×}, we must have χ(x) = 1 for all k^{×} ∩ (∏_{v} 𝔒_{v}^{×} × R^{×} × R^{×}), which is the group of units of k. Therefore we have

(4.3)  (−1)^{m_{1}+m_{2}} = 1,  sgn(ϵ^{σ_{1}})^{m_{1}}sgn(ϵ^{σ_{2}})^{m_{2}}|ϵ^{σ_{1}}|^{iv_{1}}|ϵ^{σ_{2}}|^{iv_{2}} = 1.

It is easy to see that (4.3) is a necessary and sufficient condition for χ, which is determined by χ_{∞_{1}} and χ_{∞_{2}}, to be a Hecke character of k_{A}^{×}. By (4.3), we have m_{1} = m_{2}. Put m = m_{1}. Then (4.3) is equivalent to

(4.4)  |ϵ|^{i(v_{1}−v_{2})} = sgn(N(ϵ))^{m}.

Let χ_{∗} be the associated ideal character of k. If (α), α ∈ k^{×} is a prime ideal, we have, by definition

χ_{∗}((α)) = χ((⋯, 1, ⋯, α, ⋯, 1, ⋯))
= χ((α^{−1}, ⋯, 1, ⋯, α^{−1}, ⋯)) = sgn(N(α))^{m} · (|α^{σ_{1}}|^{iv_{1}}|α^{σ_{2}}|^{iv_{2}})^{−1}.

Here (⋯, 1, ⋯, α, ⋯, 1, ⋯) ∈ k_{A}^{×} denotes the idele whose (α)-component is α and all the other components are 1. Hence we have

(4.5)  χ_{∗}((α)) = sgn(N(α))^{m}|α^{σ_{1}}|^{−iv_{1}}|α^{σ_{2}}|^{−iv_{2}}  for every α ∈ k^{×},

L(s, χ) = L(s, χ_{∗}) = ∑_{(α)} χ_{∗}((α))N((α))^{−s}. Put

R(s, χ) = |D|^{s/2}π^{−(s+m)}π^{−i(v_{1}+v_{2})/2}Γ((s + m + iv_{1})/2)
Γ((s + m + iv_{2})/2)L(s, χ),
R(s, χ^{−1}) = |D|^{s/2}π^{−(s+m)}π^{i(v_{1}+v_{2})/2}Γ((s + m − iv_{1})/2)
Γ((s + m − iv_{2})/2)L(s, χ^{−1}).

Then the functional equation is (cf. Weil [W], Langlands [LL])

(4.6)  R(s, χ) = (−1)^{m}χ_{∗}((d))R(1 − s, χ^{−1}),

where (d) denotes the different of k over Q. Since \overline{R(s, χ)} = R(s̄, χ^{−1}), we can put (4.6) in the form of (1.2):

(4.7)  R(1 − s̄, χ) = (−1)^{m}χ_{∗}((d))\overline{R(s, χ)}.

We get

(4.8)  \overline{χ_{∗}((d))^{−1/2}R(s, χ)} = (−1)^{m}χ_{∗}((d))^{−1/2}R(s, χ)  if ℜ(s) = 1/2.

Hence χ_{∗}((d))^{−1/2}R(s, χ) takes real or pure imaginary values on the critical line according as m = 0 or 1. We also note that we may assume v_{2} = 0 without losing any generality since the choice of v_{2} can be taken into account as the shift of the variable s. Then, if m = 0, we have v_{1} = −2nπ/ log ϵ with n ∈ Z by (4.4). We denote this Hecke character by χ_{n}. By (4.5), we have

(4.9)  (χ_{n})_{∗}((α)) = |α|^{2nπi/ log ϵ},  α ∈ k^{×}.

If m = 1, by (4.4), we have v_{1} = −2nπ/ log ϵ or v_{1} = −(2n + 1)π/ log ϵ with n ∈ Z according as N(ϵ) = 1 or N(ϵ) = −1. We denote this Hecke character by χ′_{n}. By (4.5), we have

(4.10)  (χ′_{n})_{∗}((α)) = \{ \begin{matrix}sgn(N(α))|α|^{2nπi/ log ϵ} if N(ϵ) = 1, \\ sgn(N(α))|α|^{(2n+1)πi/ log ϵ} if N(ϵ) = −1. \end{matrix}

As our first example, we take k = Q(\sqrt{2}). We have ϵ = \sqrt{2} + 1, (d) = (\sqrt{2})^{3}, |D| = 8. We are going to study L(s, χ_{1}) applying our summation method. For s = \frac{1}{2} + it, t = 15 and 50, the values of

R_{j} = ℜ(χ_{∗}((d))^{−1/2} exp(iϑ_{0}(t))S_{N}^{(j)}), I_{j} = ℑ(χ_{∗}((d))^{−1/2} exp(iϑ_{0}(t))S_{N}^{(j)})

are given in Tables 4.1a and in 4.1b respectively. Here χ_{∗}((d))^{−1/2} = \sqrt{2}^{−3πi/ log ϵ}, ϑ_{0}(t) = arg(8^{s/2}π^{−s}π^{2πi/ log ϵ}Γ((s − 2πi/ log ϵ)/2)Γ(s/2)). From this table, it is evident that R_{j}’s for higher j do not give good results. We can judge, from the values of |I_{j}|, R_{2} gives the best result, then R_{3}, R_{1} in this order. We must be more cautious than in §2 and §3 about the accuracy of the value e^{iϑ_{0}(t)}L(s, χ_{1}). For example, let t = 15. We empirically judge that e^{iϑ_{0}(t)}L(s, χ_{1}) = 2.17375 with error ≈ 10^{−5} from R_{2} and I_{2}. We have constructed Table 4.2 in which zeros on the critical line ℜ(s) = 1/2 are listed in the range |ℑ(s)| ≤ 30. Here, for n ≥ 1, t_{n} (resp. t_{−n}) denotes the n-th zero of L(s, χ_{1}), s = 1/2 + it on the critical line for 0 ≤ t ≤ 30 (resp. 0 ≤ −t ≤ 30).

Let us examine the Riemann hypothesis for L(s, χ_{1}) in the range 0 ≤ ℑ(s) ≤ 100. We have observed 84 sign changes of e^{iϑ_{0}(t)}L(\frac{1}{2} +it, χ_{1}) for 0 ≤ t ≤ 100. For T > 0, let N(T) denote the number of zeros of L(s, χ_{1}) counted with multiplicity in the domain

0 < ℜ(s) < 1, 0 ≤ ℑ(s) ≤ T.

Taking δ = 1/2 in (1.7) , we get

(4.11)  N(T) = π^{−1}ϑ(T) + π^{−1}Δarg(L(s, χ_{1})).

Since L(s, χ_{1}) ≠ 0 if ℜ(s) ≥ 1, Δarg(L(s, χ_{1})) equals the variation of the argument of L(s, χ_{1}) along the line segments L_{1} = [\frac{1}{2}, 1 + μ], L_{2} = [1 + μ, 1 + μ + iT], L_{3} = [1 + μ + iT, \frac{1}{2} + iT] for any μ > 0. Take T = 100 and μ = 1. We have π^{−1}ϑ(100) = 84.8864 · · ·. Hence if we can show −π < Δarg(L(s, f)) < 0, then we can conclude that N(100) = 84. For L_{1}, we divide it into 15 intervals. We observed that L(s, χ_{1}) moves from 0.3482 + 0.0712\sqrt{−1} to 0.8011 + 0.0969\sqrt{−1} keeping ℜ(L(s, χ_{1})) > 0. For L_{2}, we can show without difficulty that ℜ(L(s, χ_{1})) > 0 on L_{2}. For L_{3}, we divide it into 150 small intervals. We observed that L(s, χ_{1}) moves from 0.8159 + 0.1227\sqrt{−1} to −0.0110 − 0.0072\sqrt{−1} when s moves from 2 + 100\sqrt{−1} to \frac{1}{2} + 100\sqrt{−1}; L(s, χ_{1}) never crossed the half line ℑ(L(s, χ_{1})) = 0, ℜ(L(s, χ_{1})) ≤ 0. Hence we get N(100) = 84. The Riemann hypothesis holds and all zeros of L(s, χ_{1}) are simple zeros in this range.

By (4.11), we should have

N(100) = 84.8864 ⋯ − π^{−1} arctan(\frac{712}{3482}) − (1 − π^{−1} arctan(\frac{72}{110})) = 84.0067 · · · .

The error is about 6.7 × 10^{−3} and this is much bigger than the usual error inherent in our calculations. The reason is that L(s, χ_{1}) takes rather small value at \frac{1}{2} + 100\sqrt{−1} ; such an error can be made much smaller in the following way. We take T = 101. We have observed 86 sign changes of e^{iϑ_{0}(t)}L(\frac{1}{2} + it, χ_{1}) for 0 ≤ t ≤ 101.

We have π^{−1}ϑ(101) = 86.0881 · · ·. We divide [2 + 101\sqrt{−1}, \frac{1}{2} + 101\sqrt{−1}] into 150 small intervals. We observed that L(s, χ_{1}) moves from 0.9322 + 0.2563\sqrt{−1} to 1.7137 − 0.1288\sqrt{−1}. Hence by (4.11), we have

N(101) = 86.0881 ⋯ − π^{−1} arctan(\frac{712}{3482}) − π^{−1} arctan(\frac{1288}{17137}) = 86

with error less than 10^{−4}.

In Table 4.3, we have listed 44 examples of L(s, χ_{n}) for which we made experiments in the range 0 ≤ ℑ(s) ≤ T; N(T) denotes the number of zeros of L(s, χ_{n}) in the domain 0 < ℜ(s) < 1, 0 ≤ ℑ(s) ≤ T. We found that all zeros in the ranges of Table 4.3 lie on the critical line and are simple.

§5. Artin L-functions

Let k be the minimal splitting field of the irreducible polynomial f(X) = X^{5} − X+1 over Q. Then k ⊃ Q(\sqrt{19 · 151}) = k_{0}, k is unramified over k_{0}, Gal(k/Q) ≅ S_{5}, Gal(k/k_{0}) ≅ A_{5}. The discriminant Δ of a root of f(X) is 19 · 151. This example is due to E. Artin (cf. Lang [LG], p. 121). Let ρ be an irreducible 4-dimensional representation of S_{5} whose character χ_{ρ} is given as follows.

conjugacy class (1) (12) (123) (1234) (12)(34) (12)(345) (12345)
χ_{ρ}  4  2  1  0  0  −1  −1

Since S_{5} does not have a subgroup of index 4, ρ is not monomial. We have L(s, ρ) = ∏_{p} L_{p}(s, ρ) for ℜ(s) > 1 with the Euler p-factor L_{p}(s, ρ). We can compute L_{p}(s, ρ) as follows. First we assume that a prime number p is unramified in k, i.e., p ≠ 19,

151. Then we see easily that L_{p}(s, ρ)^{−1} equals

(1 − p^{−s})^{4}  if σ_{p}={(1)},
(1 − p^{−s})^{3}(1 + p^{−s})  if σ_{p}={(12)},
(1 − p^{−s})^{2}(1 + p^{−s} + p^{−2s})  if σ_{p}={(123)},
(1 − p^{−s})(1 + p^{−s})(1 + p^{−2s})  if σ_{p}={(1234)},
1 + p^{−s} + p^{−2s} + p^{−3s} + p^{−4s}  if σ_{p}={(12345)},
(1 − p^{−s})^{2}(1 + p^{−s})^{2}  if σ_{p}={(12)(34)},
(1 − p^{−s})(1 + p^{−s})(1 + p^{−s} + p^{−2s})  if σ_{p}={(12)(345)}.

Here σ_{p} denotes the Frobenius conjugacy class of p and {τ} denotes the conjugacy class of τ ∈ S_{5}.

Let p = 19 or 151. Let I_{𝔭} denote the inertia group of a prime factor 𝔭 of p in k. By definition, we have

(5.1)  L_{p}(s, ρ)^{−1} = det(1 − (ρ(σ_{𝔭})|V^{I_{𝔭}}) · p^{−s}).

Here V denotes the representation space of ρ, V^{I_{𝔭}} the subspace of I_{𝔭}-fixed vectors and σ_{𝔭} a Frobenius of 𝔭 which is determined modulo I_{𝔭}. Since k is unramified over k_{0}, it is obvious that |I_{𝔭}| = 2, I_{𝔭} ⊈ Gal(k/k_{0}) ≅ A_{5}. Hence we may assume that I_{𝔭} is generated by (12) choosing a suitable 𝔭 lying over p. Let D_{𝔭} denote the decomposition group of 𝔭. Then D_{𝔭} ▷ I_{𝔭} and D_{𝔭}/I_{𝔭} is generated by σ_{𝔭} mod I_{𝔭}. We have

N_{S_{5}}(I_{𝔭}) = I_{𝔭} × S_{3},

where N_{S_{5}}(I_{𝔭}) denotes the normalizer of I_{𝔭} in S_{5} and S_{3} denotes the permutation group on three letters {3, 4, 5}.

Let p = 19. Then

f(X) ≡ (X − 6)^{2}(X^{3} + 12X^{2} + 13X + 9) mod 19

is the factorization of f(X) mod p into irreducible factors in (Z/pZ)[X]. Therefore the residue field extension 𝔒_{k}/𝔭 of Z/pZ contains the cubic extension of Z/pZ, where 𝔒_{k} denotes the ring of integers of k. Hence we immediately obtain D_{𝔭}/I_{𝔭} ≅ Z/3Z and that σ_{𝔭} may be taken as (345) ∈ S_{3} . Now we find easily that

L_{p}(s, ρ)^{−1} = 1 − p^{−3s} if p = 19.

Let p = 151. Then

f(X) ≡ (X − 39)^{2}(X − 9)(X^{2} + 87X + 61) mod 151

is the factorization of f(X) mod p into irreducible factors in (Z/pZ)[X]. By a similar consideration as above, we find that D_{𝔭}/I_{𝔭} ≅ Z/2Z and that σ_{𝔭} may be taken as (34) ∈ S_{3}. We obtain

L_{p}(s, ρ)^{−1} = (1 + p^{−s})(1 − p^{−s})^{2} if p = 151.

Let f(ρ) denote the Artin conductor of ρ. We easily obtain

f(ρ) = 19 · 151.

For example, let p = 19 and 𝔭 be as above. We have shown Gal(k_{𝔭}/Q_{19}) ≅ D_{𝔭} ≅ Z/2Z × Z/3Z. We find that the restriction of ρ to D_{𝔭} splits into a direct sum of four one dimensional representations of D_{𝔭} such that three of them are unramified and one is ramified. Hence the exponent of 19 in f(ρ) is 1.

We take an isomorphism σ of k into C and let c ∈ Gal(k/Q) ≅ S_{5} be the restriction of the complex conjugation to k. Then c ∈ Gal(k/k_{0}) ≅ A_{5}. Hence c is conjugate to (12)(34) in S_{5}. Let Gal(C/R) be identified with the decomposition group ⟨c⟩ of the archimedean place of k which corresponds to σ. The restriction of ρ to Gal(C/R) splits into a direct sum of two trivial representations and two non-trivial representations. Therefore the Gamma factor to go with L(s, ρ) is given by (cf. Langlands [LL])

(π^{−s/2}Γ(s/2))^{2}(π^{−(s+1)/2}Γ((s + 1)/2))^{2}.

Put

R(s, ρ) = (19 · 151)^{s/2}π^{−2s}Γ(s/2)^{2}Γ((s + 1)/2)^{2}L(s, ρ).

Since ρ is equivalent to its contragredient, we have the functional equation

(5.2)  R(s, ρ) = κR(1 − s, ρ),

where κ = ±1 is the Artin root number attached to ρ. Let ψ be the additive character of Q_{A}/Q such that

ψ_{∞}(x) = exp(2π\sqrt{−1}x),  x ∈ Q_{∞} ≅ R,
ψ_{p}(x) = exp(−2π\sqrt{−1} Fr(x)),  x ∈ Q_{p},

where Fr denotes the fractional part of x. By a theorem of Langlands, we have

κ = ∏_{v} ϵ(\frac{1}{2}, ρ_{v}, ψ_{v})

with the ϵ-factor defined in [LL]. By the above considerations, we easily get

ϵ(\frac{1}{2}, ρ_{p}, ψ_{p}) = \{ \begin{matrix}1  if p ≠ 19, 151, \\ i  if p = 19 or 151, \end{matrix}
ϵ(\frac{1}{2}, ρ_{∞}, ψ_{∞}) = i^{2}.

Hence we obtain

(5.3)  κ = 1.

The values of

R_{j} = ℜ(exp(iϑ(t))S_{N}^{(j)}), I_{j} = ℑ(exp(iϑ(t))S_{N}^{(j)})

for L(s, ρ), s = \frac{1}{2} + it, t = 5 are given in Table 5.1. Here

ϑ(t) = arg((19 · 151)^{s/2}π^{−2s}Γ(s/2)^{2}Γ((s + 1)/2)^{2}), s = \frac{1}{2} + it.

In Table 5.2, we give the values of u_{n}, the n-th zero of L(s, ρ), s = \frac{1}{2} + iu on the critical line for 0 ≤ u ≤ 10.

§6. Estimation of errors in our calculations

The most serious defect of our method of calculation is that we do not have rigorous controle of error estimates. In previous sections, we regarded the magnitude of ℑ(e^{iϑ(t)}S_{N}^{(l)}) (resp. ℜ(e^{iϑ(t)}S_{N}^{(l)}) ) as a rough measure of errors from the true value, when e^{iϑ(t)}L(σ + it) should be real (resp. pure imaginary). In this section, we shall present several data which support this practice.

Suppose that the functional equation (1.2) for L(s) holds. Then we have e^{iϑ(t)}L(k/2 + it) ∈ R, t ∈ R where ϑ(t) = arg(κ_{1}N^{s} ∏_{i=1}^{m} Γ(b_{i}s + c_{i})), s = k/2 + it in the notation of (1.4). Take 0 < t_{1} < t_{2} so that e^{iϑ(t)} rotates on the unit circle exactly once when t moves from t_{1} to t_{2}. We expect that max_{t_{1}≤t≤t_{2}} |ℑ(e^{iϑ(t)}S_{N}^{(l)})| can be used as the measure of errors. More explicitly, it seems plausible that

(6.1)  |ℜ(e^{iϑ(t)}S_{N}^{(l)}) − L(k/2 + it)| ≤ 10 max_{t_{1}≤t≤t_{2}} |ℑ(e^{iϑ(t)}S_{N}^{(l)})|.

In examples below, we use ℜ(S_{M}^{(p)}) as a substitute for L(k/2 + it) taking large M and p (except for in Example 6), and examine the ratio of two terms in (6.1) for S_{N}^{(l)} taking relatively small N and l. The results are given in Table 6.1.

Example 1. We take the primitive form f ∈ S_{8}(Γ_{0}(2)) and consider the Lfunction L(s, f), s = 4 + it. When t moves from 97.9 to 100, e^{iϑ(t)} rotates on the unit circle approximately once. We calculated the ratio

r_{5N}^{(1)} = \frac{max_{97.9≤t≤100}(|ℜ(e^{iϑ(t)}S_{2000}^{(5N)} − e^{iϑ(t)}S_{10000}^{(35)})|)}{max_{97.9≤t≤100}(|ℑ(e^{iϑ(t)}S_{2000}^{(5N)})|)}

for 1 ≤ N ≤ 6, dividing [97.9, 100] into 21 intervals of length 0.1.

Example 2. We take g ∈ S_{9/2}(Γ_{0}(4)) as in §3. When t moves from 98.2 to 100, e^{iϑ(t)} rotates on the unit circle approximately once. We calculated the ratio

r_{5N}^{(2)} = \frac{max_{98.2≤t≤100}(|ℜ(e^{iϑ(t)}S_{2000}^{(5N)} − e^{iϑ(t)}S_{10000}^{(35)})|)}{max_{98.2≤t≤100}(|ℑ(e^{iϑ(t)}S_{2000}^{(5N)})|)}

for 1 ≤ N ≤ 6, dividing [98.2, 100] into 18 intervals of length 0.1.

Example 3. We take Δ ∈ S_{12}(SL(2, Z)) and consider the L-function L^{(3)}(s, Δ), s = 17 + it, attached to the third symmetric power representation of GL(2). When t moves from 17.5 to 20, e^{iϑ(t)} rotates on the unit circle approximately once. We calculated the ratio

r_{5N}^{(3)} = \frac{max_{17.5≤t≤20}(|ℑ(e^{iϑ(t)}S_{2000}^{(5N)} − e^{iϑ(t)}S_{10000}^{(35)})|)}{max_{17.5≤t≤20}(|ℜ(e^{iϑ(t)}S_{2000}^{(5N)})|)}

for 1 ≤ N ≤ 6, dividing [17.5, 20] into 25 intervals of length 0.1. In this example, we have normalized ϑ(t) as in §2 so that e^{iϑ(t)}L^{(3)}(17 + it, Δ) is pure imaginary.

Example 4. We consider the L-function L^{(4)}(s, Δ), s = 45/2 + it, attached to the fourth symmetric power representation of GL(2). When t moves from 7.2 to 10, e^{iϑ(t)} rotates on the unit circle approximately once. We calculated the ratio

r_{5N}^{(4)} = \frac{max_{7.2≤t≤10}(|ℜ(e^{iϑ(t)}S_{2000}^{(5N)} − e^{iϑ(t)}S_{10000}^{(35)})|)}{max_{7.2≤t≤10}(|ℑ(e^{iϑ(t)}S_{2000}^{(5N)})|)}

for 1 ≤ N ≤ 6, dividing [7.2, 10] into 28 intervals of length 0.1.

Example 5. We consider the Artin L-function treated in §5. When t moves from 4 to 5.8, e^{iϑ(t)} rotates on the unit circle approximately once. We calculated the ratio

r_{2N}^{(5)} = \frac{max_{4≤t≤5.8}(|ℜ(e^{iϑ(t)}S_{10000}^{(2N)} − e^{iϑ(t)}S_{100000}^{(10)})|)}{max_{4≤t≤5.8}(|ℑ(e^{iϑ(t)}S_{10000}^{(2N)})|)}

for 0 ≤ N ≤ 5, dividing [4, 5.8] into 18 intervals of length 0.1.

Example 6. We consider the Hecke L-function L(s, χ_{1}) for k = Q(\sqrt{2}) treated in §4. When t moves from 47.9 to 50, e^{iϑ(t)} rotates on the unit circle approximately once. We calculated the ratio

r_{N}^{(6)} = \frac{max_{47.9≤t≤50}(|ℜ(e^{iϑ(t)}S_{10000}^{(N)} − e^{iϑ(t)}S_{100000}^{(2)})|)}{max_{47.9≤t≤50}(|ℑ(e^{iϑ(t)}S_{10000}^{(N)})|)}

for 0 ≤ N ≤ 5, dividing [47.9, 50] into 21 intervals of length 0.1.

§7. A comparison with the explicit formula

We take the new form f ∈ S_{8}(Γ_{0}(2)) treated in §3. Let π be the irreducible unitary automorphic representation of GL(2, Q_{A}) which corresponds to f. We have

L_{f}(s, π) = L(s + \frac{7}{2}, f),

where L_{f}(s, π) denotes the finite part of the Jacquet-Langlands L-function attached to π. Let

L_{f}(s, π) = ∏_{p} [(1 − α_{p}p^{−s})(1 − β_{p}p^{−s})]^{−1}

be the Euler product of L_{f}(s, π). For p = 2, the Euler 2-factor degenerates so that α_{2} = −1/\sqrt{2}, β_{2} = 0. For p ≠ 2, we have |α_{p}| = |β_{p}| = 1 by the Ramanujan-Petersson conjecture proved by P. Deligne. We have the explicit formula

(7.1)

∑_{p}\begin{matrix}∑ & ′ \\ 1≤n,p^{n}≤x \end{matrix}(α_{p}^{n} + β_{p}^{n}) log p = − lim_{T →+∞} ∑_{|ℑ(ρ)|<T} \frac{x^{ρ}}{ρ} − \frac{L′_{f}(0, π)}{L_{f}(0, π)}

+ log(\frac{\sqrt{x} + 1}{\sqrt{x} − 1}) − 2x^{−1/2} − \frac{2}{3}x^{−3/2} − \frac{2}{5}x^{−5/2}

for x > 1. Here ^{∑′} means that the term (α_{p}^{n} + β_{p}^{n}) log p should be multiplied by 1/2 when p^{n} = x; ρ extends over zeros of L_{f}(s, π) such that 0 < ℜ(ρ) < 1. This formula can be shown in the usual way as in Ingham [In], p. 77–80. The last term

g(x) := log(\frac{\sqrt{x} + 1}{\sqrt{x} − 1}) − 2x^{−1/2} − \frac{2}{3}x^{−3/2} − \frac{2}{5}x^{−5/2},

which is equal to

2 ∑_{k=3}^{∞} \frac{x^{−(2k+1)/2}}{2k + 1} ,

represents the contribution of the trivial zeros of L_{f}(s, π); they are at s = −\frac{7}{2}, −\frac{9}{2}, −\frac{11}{2}, · · ·.

Now it seems very interesting to compare both sides of (7.1) numerically using the zeros of L(s, f) given in Table 3.3. We approximate

lim_{T →+∞} ∑_{|ℑ(ρ)|<T} \frac{x^{ρ}}{ρ}

by

(7.2)  h_{69}(x) := ∑_{n=1}^{69} \sqrt{x}(t_{n}^{2} + \frac{1}{4})^{−1}[cos(t_{n} log x) + 2t_{n} sin(t_{n} log x)]

with t_{n} given in Table 3.3. We have

L_{f}(0, π) = L(\frac{7}{2}, f) = 0.5942254156 ⋯, L′_{f}(0, π) = L′(\frac{7}{2}, f) = 0.1875716234 ⋯,
\frac{L′_{f}(0, π)}{L_{f}(0, π)} = 0.3156573558 · · · .

To obtain these values, we simply applied our repeated abel summation technique as before, though more rigorous evaluation could be made in this case.

In Figure 7.1, we drawed the graphs of the “step function”

∑_{p}\begin{matrix}∑ & ′ \\ 1≤n,p^{n}≤x \end{matrix}(α_{p}^{n} + β_{p}^{n}) log p

and −h_{69}(x) + g(x) − 0.3156 for 1.1 ≤ x ≤ 20. (We have used “Mathematica” to make Figure 7.1.) The coincidence seems fine.

§8. Sample programs

In this section, we shall present a few sample programs to compute L(s, f), f ∈ S_{8}(Γ_{0}(2)) (cf. §3). All programs, which are ready to be executed, are written in UBASIC created by Y. Kida. Let

f(z) = (η(z)η(2z))^{8} = ∑_{n=1}^{∞} a_{n}q^{n}.

Using Program A, we can compute a_{n} for 1 ≤ n ≤ M for any M, 1 ≤ M ≤ 10^{4}. From line 50 to 150, the coefficients A(n) in η(z) = q^{1/24} ∑_{n=1}^{∞} A(n)q^{n−1} are computed for 1 ≤ n ≤ M using Euler’s formula

η(z) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}) = q^{1/24} ∑_{n=−∞}^{∞}(−1)^{n}q^{n(3n+1)/2}.

From line 160 to 220, the coefficients B(n) in η(z)η(2z) = q^{1/8} ∑_{n=1}^{∞} B(n)q^{n−1} are computed for 1 ≤ n ≤ M by B(I) = ∑_{J+2L=I+2} A(J)A(L). From line 270 to 390, the expansion of η(z)η(2z) is raised to the eighth power; the final result will be stored in the data file “wt8 ”.

Program B computes the value of e^{iϑ_{f} (t)}L(4 + it, f) for t = 100. To save the space, this program gives the values of e^{iϑ_{f} (t)}S_{N}^{(l)} for N = 2000, 0 ≤ l ≤ 40. By point 15 command (line 20), UBASIC gives the precision to the 70-th digit. From line 300 to 580, the value of e^{iϑ_{f} (t)} will be computed and stored in the variable THE. The calculation proceeds as follows. We have

ϑ_{f}(t) = arg(2^{s/2}(2π)^{−s}Γ(s))_{s=4+it} = (\frac{1}{2} log 2 − log 2π)t + arg Γ(4 + it).

We also have

arg Γ(z) = arg Γ(z + 1) − arctan(ℑ(z)/ℜ(z)), ℜ(z) > 0, arg Γ(z) = ℑ(log Γ(z)).

By these formulas, it suffices to compute log Γ(z + 100) for z = 4 + it. We have the asymptotic expansion (cf. [WW], p. 252)

(8.1)  log Γ(z) ∼ (z − \frac{1}{2}) log z − z + \frac{1}{2} log 2π + ∑_{r=1}^{∞} \frac{(−1)^{r−1}B_{r}}{2r(2r − 1)z^{2r−1}} ,

where B_{r} denotes the r-th Bernoulli number. In (8.1), we use the terms up to r = 10. We can get an approximation of e^{iϑ_{f} (t)} which is accurate at least to the 40-th digit (cf. [WW], p. 252).

From line 660 to 740, the values of

S_{N}^{(0)} = ∑_{n=1}^{N} a_{n}n^{−s}, s_{n}^{(l)} = ∑_{n=1}^{N} s_{n}^{(l−1)}, 1 ≤ l ≤ 40, (s_{n}^{(0)} = a_{n})

are computed. From line 770 to 1020, the values of S_{N}^{(l)} are computed for 1 ≤ l ≤ 40 using (1.10) and (1.11). We use the approximation

u_{N}^{(l)} = N^{−s} ∑_{k=l}^{100}( ∑_{m=1}^{l}(−1)^{m}\binom{l}{m}m^{k})(−1)^{k} \frac{s(s + 1) ⋯ (s + k − 1)}{k!}N^{−k}.

The error from the truncation by 100 is negligible (cf. §1). The variable U(K) stands for (−1)^{k} \frac{s(s+1)···(s+k−1)}{k!}; the variable Co stands for ∑_{m=1}^{l}(−1)^{m}\binom{l}{m}m^{k}; the variable T (I) stands for

∑_{k=l}^{100}( ∑_{m=1}^{l}(−1)^{m}\binom{l}{m}m^{k})(−1)^{k} \frac{s(s + 1) ⋯ (s + k − 1)}{k!}N^{−k}.

In the line 1000, X = S_{N}^{(I)} is multiplied by e^{iϑ_{f} (t)}.

Program A

10 word 8 20 point 2 30 dim A(10000),B(10000) 40 input M 50 A1=sqrt(24*M+1) 60 M1=int((A1+1)/6) 70 A(1)=1 80 for I=1 to M1 90 I1=I−2∗int(I/2) 100 I2=1−2∗I1 110 J=int((3∗I∗I+I)/2)+1 120 A(J)=A(J)+I2 130 J=int((3∗I∗I−I)/2)+1 140 A(J)=A(J)+I2 150 next I 160 for I=1 to M 170 I1=int((I+1)/2) 180 for L=1 to I1 190 J=I−2∗L+2 200 B(I)=B(I)+A(J)∗A(L) 210 next L 220 next I 230 for I=1 to M 240 A(I)=B(I) 250 B(I)=0 260 next I 270 K=1 280 for I=1 to M 290 print K,I 300 for J=1 to I 310 B(I)=B(I)+A(J)∗A(I+1−J) 320 next J 330 next I 340 for I=1 to M 350 A(I)=B(I) 360 B(I)=0 370 next I 380 K=K+1 390 if K<4 then goto 280 400 open ”wt8” for output as ♯1 410 for I=1 to M 420 print I,A(I) 430 print ♯1,A(I) 440 next I 450 close ♯1 460 end

Program B

10 word 70 20 point 15 30 dim Bn(20),Bd(20),C(2000),T(100),U(200),Sm(50) 40 Ab=40 50 Bn(1)=1 60 Bd(1)=6 70 Bn(2)=1 80 Bd(2)=30 90 Bn(3)=1 100 Bd(3)=42 110 Bn(4)=1 120 Bd(4)=30 130 Bn(5)=5 140 Bd(5)=66 150 Bn(6)=691 160 Bd(6)=2730 170 Bn(7)=7 180 Bd(7)=6 190 Bn(8)=3617 200 Bd(8)=510 210 Bn(9)=43867 220 Bd(9)=798 230 Bn(10)=174611 240 Bd(10)=330 250 A=1/sqrt(3) 260 P=6∗atan(A) 270 S=4+100∗♯i 280 Ss=S 290 T=im(S) 300 Th=T∗(log(2)/2−log(2∗P)) 310 U=0 320 for I=1 to 100 330 S1=re(S) 340 S2=im(S) 350 if S1>S2 goto 380 360 U=U−(P/2)+atan(S1/S2) 370 goto 390 380 U=U−atan(S2/S1) 390 S=S+1 400 next I 410 Th=Th+U 420 T=im(S) 430 R=re(S)

440 X1=atan(T/R) 450 X2=(log(R∗R+T∗T))/2 460 X3=X2+X1∗♯i 470 X4=(S−(1/2))∗X3 480 Th=Th+im(X4)−T 490 K=10 500 S1=S 510 for I=1 to K 520 X1=Bn(I)/(2∗I∗(2∗I−1)∗Bd(I)∗S1) 530 Th=Th+im(X1) 540 S1=−S1∗S∗S 550 next I 560 X1=int(Th/(2∗P)) 570 X2=Th−2∗P∗X1 580 The=exp(X2∗♯i) 590 S=Ss 600 X=0 610 open ”wt8” for input as ♯1 620 for M=1 to 2000 630 input ♯1,C(M) 640 next M 650 close ♯1 660 for M=1 to 2000 670 Sm(1)=Sm(1)+C(M) 680 for I=2 to Ab 690 Sm(I)=Sm(I)+Sm(I−1) 700 next I 710 X1=log(M) 720 X2=exp(−X1∗S) 730 X=X+C(M)∗X2 740 next M 750 Z=The∗X 760 print Z 770 N=2001 780 A=log(N) 790 N1=exp(−A∗S) 800 U(1)=−S 810 for K=2 to 100 820 U(K)=−U(K−1)∗(S+K−1)/K 830 next K 840 for I=1 to Ab 850 T(I)=0 860 if I>1 then goto 890 870 X=X−Sm(1)∗N1 880 goto 1000 890 Ii=I−1 900 for K=Ii to 100 910 Co=−Ii 920 A1=−Ii 930 for L=2 to Ii 940 A1=−A1∗(Ii−L+1)/L 950 Co=Co+(A1∗(L∧K)) 960 next L 970 T(I)=T(I)+Co∗U(K)/(N∧K) 980 next K 990 X=X−Sm(I)∗N1∗T(I) 1000 Z=The∗X 1010 print I,Ss,Z 1020 next I 1030 end

§9. Conjectures

In this section, we shall discuss a few conjectures which emerged during the process of experiments: No non-trivial coincidences of zeros of two L-functions attached to two non-equivalent irreducible λ-adic representations of Gal(\overline{Q}/Q) are found. We shall use the framework of automorphic representations of GL(n, Q_{A}) to formulate this fact in more general case.

Let π be an irreducible unitary cuspidal automorphic representation of GL(n, Q_{A}). The contragredient representation π̃ to π is equivalent to the complex conjugate representation π̄ of π and we have the functional equations:

L(s, π) = ϵ(s, π)L(1 − s, π̄), L(s, π̄) = \overline{L(s̄, π)}.

Let ω_{π} be the central character of π. For s ∈ C, set ν^{s}(x) = |x|_{A}^{s}, x ∈ Q_{A}^{×} where |x|_{A} denotes the idele norm of x. We can find a t ∈ R so that ω_{π}ν^{it} is a character of Q_{A}^{×} of finite order. Since ω_{π}ν^{it} is the central character of π ⊗ (ν^{it/n} ∘ det) and L(s + \frac{it}{n}, π) = L(s, π ⊗ (ν^{it/n} ∘ det)), we may assume, without losing substantial generality, that ω_{π} is of finite order.

Conjecture 9.1. Let π_{1} and π_{2} be irreducible unitary cuspidal automorphic representations of GL(n, Q_{A}) with the central characters ω_{π_{1}} and ω_{π_{2}}. We assume that π_{1} is not equivalent to π_{2} and that ω_{π_{1}} and ω_{π_{2}} are of finite order. Then L(s, π_{1}) and L(s, π_{2}) have no common zeros in the critical strip 0 < ℜ(s) < 1 except for s = 1/2.

Remark. If we replace Q by an algebraic number field, the assertion is obviously false.

As a variant of 9.1, we can formulate a conjecture on L-functions of motives. Let E be an algebraic number field of finite degree. Let M_{1} and M_{2} be motives over Q with coefficients in E of pure weights w_{1} and w_{2}, of ranks n_{1} and n_{2} respectively. We assume w_{1} = w_{2} and put w = w_{1}. Fix an embedding σ of E into C and let L(s, M_{1}) (resp. L(s, M_{2})) be the L-function of M_{1} (resp. M_{2}) with respect to σ. For a finite place λ of E, let ρ_{i} : Gal(\overline{Q}/Q) ⟶ GL(n_{i}, E_{λ}), i = 1, 2 be the λ-adic representation obtained from the λ-adic realization of M_{i}. We assume meromorphic continuation of L(s, M_{i}), i = 1, 2 to the whole complex plane.

Conjecture 9.2. Assume that ρ_{i}, i = 1, 2 are absolutely irreducible and that ρ_{1} is not equivalent to ρ_{2} for a finite place λ of E. Then L(s, M_{1}) and L(s, M_{2}) have no common zeros in the critical strip \frac{w}{2} < ℜ(s) < \frac{w}{2} + 1 except for s = (w + 1)/2.

Remark. We understand that a pole of order k is a zero of order −k. Conjecture 9.2 implies the (usual) Artin conjecture except for s = 1/2.

Remark. The condition on ρ_{i} implies that M_{1} and M_{2} are simple motives with coefficients in \overline{Q}. D. Blasius observed that the analogous conjecture for motives over a finite field is true under the Tate conjecture (cf. Milne [M], p. 415, Proposition 2.6).

Table 2.1

N  I_{0}  R_{0}  I_{5}  R_{5}  I_{10}  R_{10}
1000 3.95 −9.3 × 10^{−2} 3.9415443 1.3 × 10^{−3} 3.9237277 3.0 × 10^{−3}
2000 4.02 3.3 × 10^{−2} 3.9248413 4.6 × 10^{−3} 3.9238190 −1.1 × 10^{−5}
4000 3.90 1.0 × 10^{−1} 3.9233635 −2.1 × 10^{−4} 3.9242209 −1.8 × 10^{−5}
6000 3.88 −3.9 × 10^{−3} 3.9239218 −3.6 × 10^{−4} 3.9242178 −3.7 × 10^{−6}
8000 3.85 −1.4 × 10^{−2} 3.9245956 6.6 × 10^{−5} 3.9241989 5.7 × 10^{−6}
10000 3.92 2.5 × 10^{−2} 3.9240769 −8.5 × 10^{−7} 3.9242054 −3.2 × 10^{−7}
N  I_{15}  R_{15}  I_{20}  R_{20}
1000 3.92361007 −1.3 × 10^{−4} 3.924178604 −4.1 × 10^{−4}
2000 3.92423322 −3.8 × 10^{−5} 3.924207205 4.9 × 10^{−6}
4000 3.92420427 1.8 × 10^{−6} 3.924203509 −1.8 × 10^{−7}
6000 3.92420338 4.6 × 10^{−7} 3.924203748 −2.9 × 10^{−8}
8000 3.92420375 −1.9 × 10^{−7} 3.924203739 7.2 × 10^{−9}
10000 3.92420370 2.4 × 10^{−8} 3.924203738 −1.0 × 10^{−9}
N  I_{25}  R_{25}  I_{30}  R_{30}
1000 3.92438069843 8.8 × 10^{−6} 3.92417001558 2.0 × 10^{−4}
2000 3.92419997465 1.2 × 10^{−6} 3.92420393248 −9.9 × 10^{−7}
4000 3.92420378031 2.6 × 10^{−8} 3.92420373301 3.0 × 10^{−9}
6000 3.92420373864 1.5 × 10^{−9} 3.92420373759 −1.7 × 10^{−10}
8000 3.92420373800 −2.7 × 10^{−10} 3.92420373818 −8.9 × 10^{−13}
10000 3.92420373814 5.8 × 10^{−11} 3.92420373812 −1.0 × 10^{−11}
N  I_{35}  R_{35}
1000 3.923939180351 4.8 × 10^{−5}
2000 3.924204540724 8.0 × 10^{−7}
4000 3.924203740852 −3.7 × 10^{−9}
6000 3.924203738244 7.9 × 10^{−11}
8000 3.924203738121 −4.4 × 10^{−12}
10000 3.924203738135 2.1 × 10^{−12}

Table 2.2

n  t_{n}  n  t_{n}  n  t_{n}
1  0  2  4.1558656464 3  5.5491219562
4  8.1117756122 5 10.8952834492 6 12.0523651120
7 13.4542992617 8 14.9275108496 9 16.3036898019
10 17.7350625418 11 18.837088412 12 20.551890978
13 21.752187480 14 22.93715924  15 23.33859940
16 23.97767239  17 25.79365179  18 27.1212236
19 27.8904904  20 28.6462091  21 30.100668
22 30.884244  23 31.730116  24 32.248613
25 33.84677  26 34.08053  27 35.12990
28 36.04356  29 36.9637  30 38.2333
31 39.1512  32 39.7944

Table 2.3

N  R_{0}  I_{0}  R_{5}  I_{5}  R_{10}  I_{10}
1000 −3.04 −7.7 × 10^{−2} −2.9558 1.4 × 10^{−2} −2.95033 9.3 × 10^{−4}
2000 −2.85 3.1 × 10^{−3} −2.9629 −3.5 × 10^{−3} −2.95651 5.4 × 10^{−4}
4000 −2.95 −9.7 × 10^{−3} −2.9588 7.3 × 10^{−4} −2.95642 −8.7 × 10^{−5}
6000 −2.95 6.3 × 10^{−2} −2.9561 −1.4 × 10^{−3} −2.95659 9.4 × 10^{−5}
8000 −2.98 6.9 × 10^{−2} −2.9558 −9.9 × 10^{−4} −2.95660 4.8 × 10^{−5}
10000 −2.97 1.8 × 10^{−2} −2.9567 −2.3 × 10^{−4} −2.95659 1.1 × 10^{−5}
N  R_{15}  I_{15}  R_{20}  I_{20}
1000 −2.956381 −2.8 × 10^{−3} −2.9591449 −4.1 × 10^{−4}
2000 −2.956387 −3.6 × 10^{−4} −2.9568292 −9.6 × 10^{−5}
4000 −2.956613 −2.3 × 10^{−5} −2.9565854 4.1 × 10^{−6}
6000 −2.956592 −3.8 × 10^{−6} −2.9565951 2.5 × 10^{−6}
8000 −2.956593 −1.7 × 10^{−6} −2.9565940 6.7 × 10^{−7}
10000 −2.956593 −2.3 × 10^{−6} −2.9565932 1.2 × 10^{−7}
N  R_{25}  I_{25}  R_{30}  I_{30}
1000 −2.9584634 3.5 × 10^{−3} −2.95201141 3.7 × 10^{−3}
2000 −2.9566395 1.9 × 10^{−4} −2.95638520 1.7 × 10^{−5}
4000 −2.9565937 −7.5 × 10^{−6} −2.95659865 2.3 × 10^{−6}
6000 −2.9565925 1.3 × 10^{−7} −2.95659358 −2.0 × 10^{−7}
8000 −2.9565932 −1.8 × 10^{−8} −2.95659347 4.3 × 10^{−8}
10000 −2.9565934 −1.0 × 10^{−7} −2.95659342 1.3 × 10^{−8}

N  R_{35}  I_{35}
1000 −2.951829903 1.3 × 10^{−4}
2000 −2.956587888 −1.9 × 10^{−4}
4000 −2.956589949 3.4 × 10^{−6}
6000 −2.956593420 1.6 × 10^{−7}
8000 −2.956593344 2.0 × 10^{−8}
10000 −2.956593405 −5.0 × 10^{−9}

Table 2.4

n  u_{n}  n  u_{n}  n  u_{n}  n  u_{n}
1  2.3864500 2  4.3752457 3  6.0435487 4  7.571907
5  8.841633 6 10.605890 7 11.437474 8 12.76622
9 13.76869 10 15.2075 11 15.6182 12 16.9663
13 18.0078  14 18.874

Table 3.1

N  R_{0}  I_{0}  R_{5}  I_{5}  R_{10}  I_{10}
1000 −1.686 8.7 × 10^{−2} −1.77582 1.6 × 10^{−2} −1.782116715 1.7 × 10^{−3}
2000 −1.835 −2.9 × 10^{−2} −1.78432 −1.4 × 10^{−3} −1.783663963 −1.9 × 10^{−5}
4000 −1.758 −6.5 × 10^{−3} −1.78364 −1.6 × 10^{−4} −1.783642384 −1.0 × 10^{−6}
6000 −1.820 −5.2 × 10^{−3} −1.78366 2.0 × 10^{−5} −1.783642948 8.6 × 10^{−8}
8000 −1.760 −1.1 × 10^{−2} −1.78364 −2.0 × 10^{−5} −1.783642816 −2.7 × 10^{−8}
10000 −1.786 8.2 × 10^{−3} −1.78363 1.1 × 10^{−5} −1.783642826 6.5 × 10^{−9}
N  R_{15}  I_{15}  R_{20}  I_{20}
1000 −1.783149679719 9.4 × 10^{−5} −1.7835293644621857 −2.4 × 10^{−5}
2000 −1.783644577360 −3.2 × 10^{−8} −1.7836428890128260 8.1 × 10^{−9}
4000 −1.783642829247 −2.2 × 10^{−9} −1.7836428272010046 −1.3 × 10^{−11}
6000 −1.783642827253 8.0 × 10^{−11} −1.7836428271539760 4.5 × 10^{−13}
8000 −1.783642827162 −6.1 × 10^{−12} −1.7836428271544534 −7.0 × 10^{−15}
10000 −1.783642827150 −1.4 × 10^{−12} −1.7836428271544125 −8.3 × 10^{−16}

N  R_{25}  I_{25}
1000 −1.7836271941181258672 −1.3 × 10^{−5}
2000 −1.7836428277732280167 1.1 × 10^{−9}
4000 −1.7836428271546296417 1.4 × 10^{−15}
6000 −1.7836428271544148349 9.0 × 10^{−16}
8000 −1.7836428271544160390 2.0 × 10^{−18}
10000 −1.7836428271544160184 −6.2 × 10^{−19}
N  R_{30}  I_{30}
1000 −1.7836430434865235424107 −3.5 × 10^{−6}
2000 −1.7836428270926205946250 5.0 × 10^{−11}
4000 −1.7836428271544153839298 1.1 × 10^{−15}
6000 −1.7836428271544160178686 −1.6 × 10^{−18}
8000 −1.7836428271544160181358 2.7 × 10^{−20}
10000 −1.7836428271544160181708 −3.8 × 10^{−22}
N  R_{35}  I_{35}
1000 −1.7836435960725853092305493 −2.3 × 10^{−7}
2000 −1.7836428271502571300252154 −3.7 × 10^{−13}
4000 −1.7836428271544160011102849 4.2 × 10^{−18}
6000 −1.7836428271544160181784305 −8.5 × 10^{−21}
8000 −1.7836428271544160181689502 1.0 × 10^{−23}
10000 −1.7836428271544160181690199 2.9 × 10^{−25}

Table 3.2

N  R_{0}  I_{0}  R_{5}  I_{5}  R_{10}  I_{10}
1000 3.43 −3.9 × 10^{−1} 3.3563 −3.5 × 10^{−1} 3.2408605 −3.5 × 10^{−1}
2000 3.23 8.9 × 10^{−2} 3.1032 6.7 × 10^{−3} 3.0945490 2.3 × 10^{−4}
4000 3.10 2.6 × 10^{−2} 3.0906 −3.1 × 10^{−4} 3.0913811 2.3 × 10^{−6}
6000 3.12 4.1 × 10^{−2} 3.0912 3.7 × 10^{−4} 3.0914333 2.3 × 10^{−6}
8000 3.07 −4.2 × 10^{−3} 3.0914 −1.7 × 10^{−4} 3.0914340 −3.9 × 10^{−7}
10000 3.12 −1.8 × 10^{−2} 3.0914 −1.3 × 10^{−4} 3.0914342 −1.6 × 10^{−7}
N  R_{15}  I_{15}  R_{20}  I_{20}
1000 3.0854245832 −3.1 × 10^{−1} 2.923318726655 −2.4 × 10^{−1}
2000 3.0919697213 −3.7 × 10^{−5} 3.091486059342 −6.5 × 10^{−5}
4000 3.0914327625 −3.9 × 10^{−8} 3.091434224098 4.0 × 10^{−8}
6000 3.0914342708 6.1 × 10^{−8} 3.091434237097 5.7 × 10^{−10}
8000 3.0914342301 −3.7 × 10^{−9} 3.091434236714 −1.0 × 10^{−11}
10000 3.0914342356 −3.6 × 10^{−10} 3.091434236735 −1.3 × 10^{−12}

N  R_{25}  I_{25}  R_{30}  I_{30}
1000 2.80762049069122 −1.1 × 10^{−1} 2.78668118045678635 7.4 × 10^{−2}
2000 3.09143198592104 −2.2 × 10^{−5} 3.09143105396801472 −3.5 × 10^{−6}
4000 3.09143423720919 2.4 × 10^{−9} 3.09143423678556380 3.9 × 10^{−11}
6000 3.09143423674457 −5.1 × 10^{−12} 3.09143423673870285 −1.7 × 10^{−13}
8000 3.09143423673861 2.3 × 10^{−13} 3.09143423673865144 1.9 × 10^{−15}
10000 3.09143423673865 1.0 × 10^{−14} 3.09143423673865098 6.2 × 10^{−17}
N  R_{35}  I_{35}
1000 2.91398635132232829470 2.7 × 10^{−1}
2000 3.09143341359018088974 1.4 × 10^{−7}
4000 3.09143423673979921383 −2.4 × 10^{−12}
6000 3.09143423673865010310 −1.1 × 10^{−15}
8000 3.09143423673865095777 −1.9 × 10^{−18}
10000 3.09143423673865095240 −1.3 × 10^{−20}

Table 3.3

\begin{matrix}n & t_{n} & n & t_{n} & n & t_{n} \\ 1 & 8.2720409199 & 2 & 11.3959869930 & 3 & 14.8616932015 \\ 4 & 17.1783243050 & 5 & 19.2124566315 & 6 & 20.8274294554 \\ 7 & 23.4659374198 & 8 & 25.2726883522 & 9 & 27.0035774491 \\ 10 & 28.1569222690 & 11 & 30.2145623343 & 12 & 31.6193141164 \\ 13 & 33.7856279775 & 14 & 34.9435854723 & 15 & 36.5559515067 \\ 16 & 37.6356026748 & 17 & 39.1608229256 & 18 & 40.6589300308 \\ 19 & 42.8804581030 & 20 & 43.2736012304 & 21 & 44.9765395474 \\ 22 & 46.4176568046 & 23 & 47.2517710599 & 24 & 48.7821808287 \\ 25 & 50.3519022325 & 26 & 51.5688981695 & 27 & 53.1356287828 \\ 28 & 54.0717837181 & 29 & 55.0990003336 & 30 & 56.4089955139 \\ 31 & 57.5391214415 & 32 & 59.1986375433 & 33 & 60.1739007171 \\ 34 & 61.6441827270 & 35 & 62.8146545420 & 36 & 63.4247884022 \\ 37 & 65.1023702197 & 38 & 66.0180646898 & 39 & 66.8050237006 \\ 40 & 68.6802278238 & 41 & 69.8132058342 & 42 & 70.7502185552 \\ 43 & 71.9861530156 & 44 & 72.7927328082 & 45 & 74.1137296216 \\ 46 & 74.8761895173 & 47 & 76.2796967025 & 48 & 77.4608764665 \\ 49 & 78.7319975717 & 50 & 79.5372511477 & 51 & 80.8499015926 \\ 52 & 81.9286308045 & 53 & 82.6995529553 & 54 & 83.4681179192 \\ 55 & 85.2402769759 & 56 & 85.6802121224 & 57 & 87.2830188249 \\ 58 & 88.5094955323 & 59 & 89.2377130355 & 60 & 90.0534073382 \\ 61 & 91.4472572430 & 62 & 92.0496894589 & 63 & 93.3566370961 \\ 64 & 94.2221147468 & 65 & 95.3044565474 & 66 & 96.6527715250 \\ 67 & 97.7264314003 & 68 & 98.4244540180 & 69 & 99.4730638315 \end{matrix}

\begin{matrix}n & u_{n} & n & u_{n} & n & u_{n} \\ 1 & 12.9399446108 & 2 & 15.1248640287 & 3 & 17.2775088490 \\ 4 & 21.9119654118 & 5 & 23.7124474310 & 6 & 27.6868648494 \\ 7 & 29.1470584255 & 8 & 31.1315265360 & 9 & 31.9862854000 \\ 10 & 33.6323734231 & 11 & 35.7361264638 & 12 & 38.1008875317 \\ 13 & 39.9548075690 & 14 & 41.3629251312 & 15 & 43.0030848131 \\ 16 & 43.7924301232 & 17 & 49.3874980802 & 18 & 50.3892911690 \\ 19 & 51.9883497256 & 20 & 53.3610715851 & 21 & 55.5058736308 \\ 22 & 57.1306190068 & 23 & 58.5145765119 & 24 & 59.2810504632 \\ 25 & 60.8114113807 & 26 & 61.7177742037 & 27 & 62.3299217969 \\ 28 & 65.1200148215 & 29 & 66.3871768599 & 30 & 67.7658152255 \\ 31 & 68.5636482576 & 32 & 70.0994795387 & 33 & 71.9139076205 \\ 34 & 73.4598285562 & 35 & 74.4219698604 & 36 & 75.9259426071 \\ 37 & 76.7219513797 & 38 & 80.2370604179 & 39 & 80.9795116625 \\ 40 & 82.2141987387 & 41 & 84.1686266809 & 42 & 85.4558525934 \\ 43 & 86.4596989555 & 44 & 87.5017176801 & 45 & 88.7989284271 \\ 46 & 90.8930274655 & 47 & 91.6094970880 & 48 & 93.0648834307 \\ 49 & 93.8455295242 & 50 & 94.4090447654 & 51 & 95.8637476000 \\ 52 & 96.5025102751 & 53 & 97.8219970593 & 54 & 98.9086789539 \end{matrix}

Table 3.5

\begin{matrix}n & ρ_{n} & n & ρ_{n} \\ 1 & 3.2308208282 + 8.9496290911 i & 2 & 3.0144204971 + 19.1670355895 i \\ 3 & 3.1880664988 + 26.3033849287 i & 4 & 3.1549639910 + 36.6242398231 i \\ 5 & 2.7150409653 + 45.1719799932 i & 6 & 2.4938210677 + 47.5816502442 i \\ 7 & 3.3624175212 + 54.4320525502 i & 8 & 2.9077749773 + 64.2513434784 i \\ 9 & 3.1556119321 + 71.9344926377 i & 10 & 2.4066868777 + 78.1144688947 i \\ 11 & 2.9102154501 + 82.3890698796 i & 12 & 2.8870784016 + 89.7875787849 i \\ 13 & 3.3672596002 + 99.9194124003 i & 14 & 2.7073645379 + 107.1592978688 i \\ 15 & 2.7721770492 + 110.2613188689 i & 16 & 3.1645227049 + 117.3896482956 i \\ 17 & 2.5547542302 + 126.5001914198 i & 18 & 2.8669588498 + 128.2070045099 i \\ 19 & 3.1444059195 + 135.3354942155 i & 20 & 3.3550056102 + 145.3938901873 i \end{matrix}

n  u_{n}^{+}  n  u_{n}^{+}  n  u_{n}^{+}
1 5.6185671952 2 9.3587692608 3 12.0264936925
4 13.7962357292 5 16.0894994874 6 17.7847280762
7 19.4575970308 8 21.8594962784 9 22.5758373316
10 24.1962387103 11 26.0225517432 12 27.3617234087
13 29.1281915898 14 29.9334498107 15 31.7848563053
16 33.0414393853 17 33.9453293541 18 35.7534640159
19 36.6630191145 20 38.2010379041 21 39.7144586843
22 40.6779144658 23 41.3507131813 24 43.0237415354
25 44.5137568744 26 45.2712575072 27 46.4744214908
28 47.8221666146 29 49.0105055856 30 49.9881789218
31 51.5155718913 32 51.6631984921 33 53.5375246718
34 54.4042145877 35 55.5750957955 36 58.8270009777
37 59.7418610167 38 61.2006909669 39 61.7440856002
40 63.2789136582 41 64.1010371646 42 64.8912913186
43 66.3286609040 44 67.5430514279 45 70.1770012608
46 71.4492522675 47 72.3027663172 48 73.1469069994
49 74.5424808567 50 75.2004527728 51 76.3094730233
52 77.0682809138 53 78.2931224271 54 79.2581319133
55 80.0330330946 56 81.1315223512 57 81.9704731391
58 83.0719604785 59 84.1964169499 60 85.2011952740
61 85.7778751490 62 86.9407183317 63 87.4479355411
64 88.7059787721 65 89.9368381341 66 90.6181312264
67 91.4583733416 68 92.7393858316 69 93.0881396596
70 94.5091774704 71 95.4769845208 72 96.1116129723
73 96.9317027531 74 97.8372552255 75 99.1010053953
76 99.8881597950
n  u_{n}^{−}  n  u_{n}^{−}  n  u_{n}^{−}
1 0  2 24.4022873037 3 26.6418851276
4 29.3670678246 5 33.5747954436 6 35.2863556538
7 38.5418813017 8 40.0447318001 9 44.8539372315
10 46.8484465576 11 50.0699839799 12 52.1256566323
13 54.2746979473 14 55.4033486176 15 56.5807632829
16 58.8640673277 17 60.9985910184 18 63.8456398387
19 65.1978869599 20 71.4691922483 21 72.8978373808
22 75.0463737748 23 76.6579317410 24 78.6419254744
25 80.0462305996 26 83.8815457436 27 85.2027989008
28 86.6795992346 29 88.7015447955 30 90.5599843766
31 92.9081333908 32 94.5315054431 33 96.4777165784
34 97.7100184923 35 99.3454122963

n  ρ_{n}^{+}  n  ρ_{n}^{+}
1 3.3591319232 + 57.3250633340 i 2 3.5001503209 + 68.5679322965 i
n  ρ_{n}^{−}  n  ρ_{n}^{−}
1 5.7692647648 + 8.9956889852 i 2 4.8476735625 + 14.0858508094 i
3 5.3846794177 + 18.2757545274 i 4 4.2067408135 + 20.6821222248 i
5 5.3861276711 + 26.6587658619 i 6 4.5402410961 + 31.5997171480 i
7 5.7862146670 + 36.6524415925 i 8 4.3281324452 + 41.9810464020 i
9 5.2171816339 + 44.9332286844 i 10 4.7664043254 + 48.8240853600 i
11 5.6527005509 + 54.2712676115 i 12 4.6702375442 + 60.1227186672 i
13 5.2007749392 + 63.8919812818 i 14 3.7792244502 + 66.9023711945 i
15 3.5790185110 + 69.3185823254 i 16 5.7686816361 + 72.2279984677 i
17 4.6599017691 + 76.8392451729 i 18 5.3421407237 + 81.9810448650 i
19 3.3026675429 + 82.2734272886 i 20 3.9986981918 + 88.8368300771 i
21 4.9626158230 + 90.6464501774 i 22 4.7401362618 + 94.8547536615 i
23 5.8412001878 + 99.7795574645 i

Table 4.1a

N  R_{0}  I_{0}  R_{1}  I_{1}  R_{2}  I_{2}
1000 2.22 6.3 × 10^{−3} 2.1777 −1.1 × 10^{−2} 2.173977 2.6 × 10^{−4}
5000 2.14 −2.2 × 10^{−2} 2.1700 3.7 × 10^{−3} 2.173544 −1.2 × 10^{−4}
10000 2.18 −1.3 × 10^{−2} 2.1753 1.1 × 10^{−3} 2.173833 −1.0 × 10^{−4}
30000 2.19 −2.2 × 10^{−2} 2.1729 −6.9 × 10^{−4} 2.173759 −1.0 × 10^{−5}
100000 2.17 1.2 × 10^{−2} 2.1735 1.0 × 10^{−4} 2.173747 −4.7 × 10^{−6}
N  R_{3}  I_{3}  R_{4}  I_{4}  R_{5}  I_{5}
1000 2.167862 −6.2 × 10^{−3} 2.20484 −5.4 × 10^{−2} 2.49186 7.1 × 10^{−2}
5000 2.173880 −6.8 × 10^{−4} 2.17935 −1.6 × 10^{−4} 2.18302 2.7 × 10^{−2}
10000 2.173937 1.8 × 10^{−4} 2.17267 1.6 × 10^{−3} 2.16380 −2.8 × 10^{−3}
30000 2.173700 −2.3 × 10^{−6} 2.17367 −3.7 × 10^{−4} 2.17551 −9.3 × 10^{−4}
100000 2.173745 −5.9 × 10^{−6} 2.17378 −5.1 × 10^{−5} 2.17406 9.7 × 10^{−5}

N  R_{6}  I_{6}  R_{7}  I_{7}  R_{8}  I_{8}
1000 2.352  1.2  −1.442  2.2  −7.975  −6.1
5000 2.082 7.3 × 10^{−2} 1.835 −1.7 × 10^{−1} 2.195  −1.0
10000 2.170 −4.1 × 10^{−2} 2.293 −6.2 × 10^{−2} 2.480  2.2 × 10^{−1}
30000 2.179 5.3 × 10^{−3} 2.165 2.4 × 10^{−2} 2.101 1.0 × 10^{−2}
100000 2.173 1.3 × 10^{−3} 2.169  1.8 × 10^{−3} 2.164 −7.2 × 10^{−3}

Table 4.1b

N  R_{0}  I_{0}  R_{1}  I_{1}  R_{2}  I_{2}
1000 3.263 8.3 × 10^{−2} 3.3048 6.1 × 10^{−2} 3.28176 2.6 × 10^{−2}
5000 3.305 −1.1 × 10^{−2} 3.2708 −1.8 × 10^{−2} 3.26879 −8.6 × 10^{−4}
10000 3.245 6.4 × 10^{−3} 3.2612 −4.8 × 10^{−3} 3.26487 6.8 × 10^{−4}
30000 3.263 2.9 × 10^{−2} 3.2697 9.0 × 10^{−4} 3.26616 1.7 × 10^{−4}
100000 3.276 −8.2 × 10^{−3} 3.2665 5.8 × 10^{−4} 3.26610 4.9 × 10^{−5}
N  R_{3}  I_{3}  R_{4}  I_{4}  R_{5}  I_{5}
1000 3.38022 2.5 × 10^{−2} 3.524  2.2  −33.06  8.3
5000 3.26878 6.2 × 10^{−3} 3.080 7.2 × 10^{−2}  1.75  −2.9
10000 3.26486 −2.7 × 10^{−3} 3.329 −3.1 × 10^{−2}  3.88  1.0
30000 3.26672 −1.6 × 10^{−4} 3.271 1.2 × 10^{−2}  3.06  1.0 × 10^{−1}
100000 3.26609 6.8 × 10^{−5} 3.264 −1.3 × 10^{−3}  3.28  −3.1 × 10^{−2}

Table 4.2

n  t_{n}  n  t_{n}  n  t_{n}  n  t_{n}
1 10.2562  2 13.6866  3 15.9599  4 17.038
5 19.026  6 20.017  7 22.472  8 23.745
9 25.351  10 26.229  11 27.561  12 28.847
13 29.986
−1 −3.12740 −2 −6.5577 −3 −8.8310 −4 −9.9092
−5 −11.8976 −6 −12.888 −7 −15.343 −8 −16.616
−9 −18.222 −10 −19.101 −11 −20.433 −12 −21.718
−13 −22.857 −14 −24.859 −15 −25.892 −16 −26.850
−17 −28.418 −18 −28.927

k  n T N(T)  k  n T N(T)
Q(\sqrt{2}) 1 100 84 Q(\sqrt{2}) −1 100 93
Q(\sqrt{2}) 2 100 82 Q(\sqrt{2}) −2 100 96
Q(\sqrt{2}) 3 100 81 Q(\sqrt{2}) −3 100 98
Q(\sqrt{2}) 4 100 80 Q(\sqrt{2}) −4 100 100
Q(\sqrt{2}) 5 100 78 Q(\sqrt{2}) −5 100 101
Q(\sqrt{2}) 10 100 79 Q(\sqrt{2}) −10 100 108
Q(\sqrt{5}) 1 100 75 Q(\sqrt{5}) −1 100 88
Q(\sqrt{5}) 2 100 72 Q(\sqrt{5}) −2 100 92
Q(\sqrt{5}) 3 100 71 Q(\sqrt{5}) −3 100 95
Q(\sqrt{5}) 4 100 71 Q(\sqrt{5}) −4 100 97
Q(\sqrt{5}) 5 100 71 Q(\sqrt{5}) −5 100 99
Q(\sqrt{5}) 10 100 93 Q(\sqrt{5}) −10 100 106
Q(\sqrt{19}) 1 50 51 Q(\sqrt{19}) −1 50 52
Q(\sqrt{19}) 2 50 50 Q(\sqrt{19}) −2 50 53
Q(\sqrt{29}) 1 100 107 Q(\sqrt{29}) −1 100 112
Q(\sqrt{29}) 2 100 105 Q(\sqrt{29}) −2 100 114
Q(\sqrt{29}) 3 100 104 Q(\sqrt{29}) −3 100 116
Q(\sqrt{29}) 4 100 102 Q(\sqrt{29}) −4 100 117
Q(\sqrt{29}) 5 100 102 Q(\sqrt{29}) −5 100 118
Q(\sqrt{29}) 10 100 99 Q(\sqrt{29}) −10 100 123
Q(\sqrt{31}) 1 40 41 Q(\sqrt{31}) −1 40 42
Q(\sqrt{67}) 1 30 32 Q(\sqrt{67}) −1 30 33

Table 5.1

N  R_{0}  I_{0}  R_{2}  I_{2}  R_{4}  I_{4}
1000 3.09 5.3 × 10^{−1} 3.334 4.5 × 10^{−1} 3.483 4.9 × 10^{−1}
5000 3.31 −5.5 × 10^{−2} 3.317 −2.4 × 10^{−2} 3.335 −3.2 × 10^{−3}
10000 3.42 1.5 × 10^{−1} 3.417 −7.8 × 10^{−3} 3.389 −1.3 × 10^{−2}
30000 3.45 −1.4 × 10^{−1} 3.385 −2.3 × 10^{−2} 3.380 −6.2 × 10^{−3}
100000 3.25 −4.8 × 10^{−2} 3.375 1.6 × 10^{−3} 3.383 1.2 × 10^{−3}
N  R_{6}  I_{6}  R_{8}  I_{8}  R_{10}  I_{10}
1000 3.9634 6.4 × 10^{−1} 4.03999 6.2 × 10^{−1} 4.792707 5.2 × 10^{−1}
5000 3.3419 3.6 × 10^{−2} 3.38817 4.1 × 10^{−2} 3.400866 6.3 × 10^{−2}
10000 3.3825 −1.0 × 10^{−2} 3.37107 −4.5 × 10^{−3} 3.377550 4.8 × 10^{−3}
30000 3.3803 1.9 × 10^{−4} 3.38367 1.4 × 10^{−3} 3.383935 3.6 × 10^{−4}
100000 3.3839 1.2 × 10^{−4} 3.38369 −8.5 × 10^{−5} 3.383657 −8.2 × 10^{−6}

Table 5.2

n  u_{n}  n  u_{n}  n  u_{n} n  u_{n} n  u_{n}
1 2.79373 2 4.0887 3 5.362 4 5.887 5 7.03
6 7.46  7 7.90  8 8.8  9 9.6

Table 6.1

\begin{matrix}N & r_{N}^{(1)} & r_{N}^{(2)} & r_{N}^{(3)} & r_{N}^{(4)} & N & r_{N}^{(5)} & N & r_{N}^{(6)} \\ 5 & 0.96 & 0.98 & 0.93 & 0.92 & 0 & 1.03 & 0 & 0.98 \\ 10 & 0.95 & 1.05 & 1.12 & 0.84 & 2 & 0.82 & 1 & 0.99 \\ 15 & 1.07 & 1.06 & 0.85 & 1.03 & 4 & 0.81 & 2 & 1.01 \\ 20 & 1.07 & 1.09 & 1.23 & 1.07 & 6 & 1.19 & 3 & 0.99 \\ 25 & 1.06 & 0.90 & 1.25 & 0.87 & 8 & 1.39 & 4 & 1.01 \\ 30 & 0.88 & 0.88 & 0.80 & 1.22 & 10 & 0.67 & 5 & 0.94 \end{matrix}

Figure 7.1

36

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