| L(s) = 1 | − 3.49e13·2-s + 3.81e21·3-s − 1.25e27·4-s + 4.98e31·5-s − 1.33e35·6-s + 4.56e38·7-s + 1.30e41·8-s − 1.16e43·9-s − 1.74e45·10-s + 4.07e47·11-s − 4.77e48·12-s − 8.01e49·13-s − 1.59e52·14-s + 1.90e53·15-s − 1.46e54·16-s + 8.73e55·17-s + 4.05e56·18-s + 1.44e58·19-s − 6.23e58·20-s + 1.74e60·21-s − 1.42e61·22-s − 7.95e61·23-s + 4.98e62·24-s − 1.55e63·25-s + 2.80e63·26-s − 1.44e65·27-s − 5.71e65·28-s + ⋯ |
| L(s) = 1 | − 0.703·2-s + 0.746·3-s − 0.505·4-s + 0.784·5-s − 0.524·6-s + 1.61·7-s + 1.05·8-s − 0.443·9-s − 0.551·10-s + 1.68·11-s − 0.377·12-s − 0.165·13-s − 1.13·14-s + 0.585·15-s − 0.238·16-s + 0.903·17-s + 0.311·18-s + 0.950·19-s − 0.396·20-s + 1.20·21-s − 1.18·22-s − 0.874·23-s + 0.790·24-s − 0.385·25-s + 0.116·26-s − 1.07·27-s − 0.815·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(92-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+91/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(46)\) |
\(\approx\) |
\(2.573126113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.573126113\) |
| \(L(\frac{93}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 3.49e13T + 2.47e27T^{2} \) |
| 3 | \( 1 - 3.81e21T + 2.61e43T^{2} \) |
| 5 | \( 1 - 4.98e31T + 4.03e63T^{2} \) |
| 7 | \( 1 - 4.56e38T + 8.01e76T^{2} \) |
| 11 | \( 1 - 4.07e47T + 5.84e94T^{2} \) |
| 13 | \( 1 + 8.01e49T + 2.33e101T^{2} \) |
| 17 | \( 1 - 8.73e55T + 9.35e111T^{2} \) |
| 19 | \( 1 - 1.44e58T + 2.32e116T^{2} \) |
| 23 | \( 1 + 7.95e61T + 8.26e123T^{2} \) |
| 29 | \( 1 + 2.07e66T + 1.19e133T^{2} \) |
| 31 | \( 1 - 4.78e67T + 5.17e135T^{2} \) |
| 37 | \( 1 + 2.39e71T + 5.08e142T^{2} \) |
| 41 | \( 1 - 7.71e72T + 5.79e146T^{2} \) |
| 43 | \( 1 - 3.25e74T + 4.42e148T^{2} \) |
| 47 | \( 1 + 5.86e75T + 1.44e152T^{2} \) |
| 53 | \( 1 + 7.91e76T + 8.11e156T^{2} \) |
| 59 | \( 1 - 4.25e80T + 1.40e161T^{2} \) |
| 61 | \( 1 - 2.77e81T + 2.91e162T^{2} \) |
| 67 | \( 1 + 1.96e83T + 1.48e166T^{2} \) |
| 71 | \( 1 - 2.44e84T + 2.91e168T^{2} \) |
| 73 | \( 1 - 2.66e84T + 3.65e169T^{2} \) |
| 79 | \( 1 + 5.05e85T + 4.83e172T^{2} \) |
| 83 | \( 1 + 2.95e87T + 4.32e174T^{2} \) |
| 89 | \( 1 - 9.63e87T + 2.48e177T^{2} \) |
| 97 | \( 1 + 2.03e90T + 6.25e180T^{2} \) |
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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
−14.31537090314289610024672408048, −14.06818127176497906564974584065, −11.61693779999474139973759356553, −9.780752129847364013122202335729, −8.766724610904073979657623788462, −7.71119576384368311892625942927, −5.47315257991503690444654948454, −3.93551327190529241826278153527, −1.93819221403499507543223120512, −1.08198792998400779060067216977,
1.08198792998400779060067216977, 1.93819221403499507543223120512, 3.93551327190529241826278153527, 5.47315257991503690444654948454, 7.71119576384368311892625942927, 8.766724610904073979657623788462, 9.780752129847364013122202335729, 11.61693779999474139973759356553, 14.06818127176497906564974584065, 14.31537090314289610024672408048
