| L(s) = 1 | + (−11.0 + 19.0i)2-s + (11.3 + 19.5i)3-s + (−178. − 308. i)4-s + (50.3 + 87.1i)5-s − 497.·6-s + (−276. − 478. i)7-s + 5.01e3·8-s + (837. − 1.45e3i)9-s − 2.21e3·10-s + 5.90e3·11-s + (4.02e3 − 6.97e3i)12-s + (−4.09e3 − 7.09e3i)13-s + 1.21e4·14-s + (−1.13e3 + 1.97e3i)15-s + (−3.24e4 + 5.61e4i)16-s + (−1.06e4 + 1.84e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.972 + 1.68i)2-s + (0.241 + 0.419i)3-s + (−1.39 − 2.40i)4-s + (0.180 + 0.311i)5-s − 0.941·6-s + (−0.304 − 0.527i)7-s + 3.46·8-s + (0.382 − 0.663i)9-s − 0.700·10-s + 1.33·11-s + (0.673 − 1.16i)12-s + (−0.516 − 0.895i)13-s + 1.18·14-s + (−0.0871 + 0.150i)15-s + (−1.97 + 3.42i)16-s + (−0.527 + 0.913i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.510 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.898154 + 0.511417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.898154 + 0.511417i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + (-3.04e5 - 4.87e4i)T \) |
| good | 2 | \( 1 + (11.0 - 19.0i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (-11.3 - 19.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-50.3 - 87.1i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (276. + 478. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 - 5.90e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (4.09e3 + 7.09e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (1.06e4 - 1.84e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.63e4 + 2.83e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 - 4.52e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.10e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.00e4T + 2.75e10T^{2} \) |
| 41 | \( 1 + (-1.38e5 - 2.40e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 - 8.01e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.57e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-9.96e5 + 1.72e6i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (4.99e5 - 8.65e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-3.23e5 - 5.59e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (4.60e5 + 7.96e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-6.30e4 - 1.09e5i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + 3.88e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (1.75e6 + 3.03e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-4.73e6 + 8.19e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (5.74e6 - 9.95e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.54e7T + 8.07e13T^{2} \) |
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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
−14.97396890890171928928017146712, −14.71866156295740072401695164478, −13.16969603353794875428982245615, −10.62467931507071965425831592142, −9.611858885740001145402979648295, −8.682455488842579756099501009261, −7.09385045049158494174705926145, −6.26045392280145617655599218876, −4.35098845170696330853558665269, −0.74537183768581327795378989081,
1.28327413241279366533197059874, 2.46064417327063508637551286876, 4.29541466580111994096685262551, 7.30621620021644301818240909422, 8.892361832698854112122772776348, 9.487374085785482559002513478836, 11.01050937793017089761181476750, 12.09318199638545216365968228513, 12.92918585990453519697495391777, 14.10824233101935047986164684291
