| L(s) = 1 | + (−14.2 + 5.18i)2-s + (85.5 + 31.1i)3-s + (78.1 − 65.5i)4-s + (80.5 + 457. i)5-s − 1.38e3·6-s + (115. + 654. i)7-s + (197. − 341. i)8-s + (4.67e3 + 3.92e3i)9-s + (−3.51e3 − 6.09e3i)10-s + (1.82e3 − 3.16e3i)11-s + (8.73e3 − 3.17e3i)12-s + (5.30e3 − 4.45e3i)13-s + (−5.03e3 − 8.72e3i)14-s + (−7.33e3 + 4.16e4i)15-s + (−3.30e3 + 1.87e4i)16-s + (−1.06e4 − 8.96e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.25 + 0.458i)2-s + (1.83 + 0.666i)3-s + (0.610 − 0.512i)4-s + (0.288 + 1.63i)5-s − 2.61·6-s + (0.127 + 0.720i)7-s + (0.136 − 0.235i)8-s + (2.13 + 1.79i)9-s + (−1.11 − 1.92i)10-s + (0.414 − 0.717i)11-s + (1.45 − 0.530i)12-s + (0.670 − 0.562i)13-s + (−0.490 − 0.849i)14-s + (−0.561 + 3.18i)15-s + (−0.201 + 1.14i)16-s + (−0.527 − 0.442i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.750860 + 1.69388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.750860 + 1.69388i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (2.80e5 + 1.27e5i)T \) |
| good | 2 | \( 1 + (14.2 - 5.18i)T + (98.0 - 82.2i)T^{2} \) |
| 3 | \( 1 + (-85.5 - 31.1i)T + (1.67e3 + 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-80.5 - 457. i)T + (-7.34e4 + 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-115. - 654. i)T + (-7.73e5 + 2.81e5i)T^{2} \) |
| 11 | \( 1 + (-1.82e3 + 3.16e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-5.30e3 + 4.45e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (1.06e4 + 8.96e3i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 19 | \( 1 + (-2.68e3 - 976. i)T + (6.84e8 + 5.74e8i)T^{2} \) |
| 23 | \( 1 + (3.63e4 + 6.29e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-2.29e4 + 3.97e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 3.36e4T + 2.75e10T^{2} \) |
| 41 | \( 1 + (4.26e5 - 3.57e5i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 - 4.87e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.01e5 + 3.48e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-4.85e4 + 2.75e5i)T + (-1.10e12 - 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-3.68e5 + 2.08e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.47e6 + 1.23e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-2.42e5 - 1.37e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (2.82e6 + 1.02e6i)T + (6.96e12 + 5.84e12i)T^{2} \) |
| 73 | \( 1 - 2.45e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (1.03e6 + 5.89e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (-2.83e6 - 2.38e6i)T + (4.71e12 + 2.67e13i)T^{2} \) |
| 89 | \( 1 + (1.87e6 - 1.06e7i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (-3.44e6 - 5.96e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.34610213015399730954438588891, −14.44554083089121134600396592603, −13.51271418493730589520326351625, −10.80744367800943567454440520382, −9.943179837518585174720785817037, −8.845436971473023690882624240318, −8.029709434861598032406446059527, −6.66952612172978210452929287469, −3.55149141390259482876971224724, −2.28476354915795109430398186667,
1.16494235567948265112599716281, 1.82290700576689788202540302471, 4.15461896022643070469378584523, 7.31163704510216058987976581696, 8.475300246802255802619768346565, 9.031813304759562760870673515553, 9.962528248775426274665953470694, 12.12967903415535547623296293374, 13.35111564460094625756515763735, 14.05363232355245426381725616907
