| L(s) = 1 | + 7.65e13·2-s − 2.65e21·3-s + 3.37e27·4-s − 4.45e31·5-s − 2.03e35·6-s + 2.84e38·7-s + 6.89e40·8-s − 1.91e43·9-s − 3.40e45·10-s + 9.72e46·11-s − 8.97e48·12-s + 6.30e50·13-s + 2.17e52·14-s + 1.18e53·15-s − 3.08e54·16-s − 5.77e55·17-s − 1.46e57·18-s + 1.72e58·19-s − 1.50e59·20-s − 7.55e59·21-s + 7.44e60·22-s + 1.13e62·23-s − 1.83e62·24-s − 2.05e63·25-s + 4.82e64·26-s + 1.20e65·27-s + 9.59e65·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.519·3-s + 1.36·4-s − 0.701·5-s − 0.798·6-s + 1.00·7-s + 0.559·8-s − 0.730·9-s − 1.07·10-s + 0.402·11-s − 0.708·12-s + 1.30·13-s + 1.54·14-s + 0.364·15-s − 0.503·16-s − 0.597·17-s − 1.12·18-s + 1.12·19-s − 0.956·20-s − 0.521·21-s + 0.618·22-s + 1.25·23-s − 0.290·24-s − 0.508·25-s + 2.00·26-s + 0.898·27-s + 1.36·28-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
| \(L(46)\) |
\(\approx\) |
\(4.248639536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.248639536\) |
| \(L(\frac{93}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 7.65e13T + 2.47e27T^{2} \) |
| 3 | \( 1 + 2.65e21T + 2.61e43T^{2} \) |
| 5 | \( 1 + 4.45e31T + 4.03e63T^{2} \) |
| 7 | \( 1 - 2.84e38T + 8.01e76T^{2} \) |
| 11 | \( 1 - 9.72e46T + 5.84e94T^{2} \) |
| 13 | \( 1 - 6.30e50T + 2.33e101T^{2} \) |
| 17 | \( 1 + 5.77e55T + 9.35e111T^{2} \) |
| 19 | \( 1 - 1.72e58T + 2.32e116T^{2} \) |
| 23 | \( 1 - 1.13e62T + 8.26e123T^{2} \) |
| 29 | \( 1 - 5.78e66T + 1.19e133T^{2} \) |
| 31 | \( 1 - 5.69e67T + 5.17e135T^{2} \) |
| 37 | \( 1 - 4.23e71T + 5.08e142T^{2} \) |
| 41 | \( 1 + 2.50e72T + 5.79e146T^{2} \) |
| 43 | \( 1 + 1.35e74T + 4.42e148T^{2} \) |
| 47 | \( 1 + 1.35e76T + 1.44e152T^{2} \) |
| 53 | \( 1 - 4.17e78T + 8.11e156T^{2} \) |
| 59 | \( 1 - 3.62e80T + 1.40e161T^{2} \) |
| 61 | \( 1 - 9.25e80T + 2.91e162T^{2} \) |
| 67 | \( 1 - 1.21e83T + 1.48e166T^{2} \) |
| 71 | \( 1 + 9.39e82T + 2.91e168T^{2} \) |
| 73 | \( 1 + 5.75e84T + 3.65e169T^{2} \) |
| 79 | \( 1 + 1.40e86T + 4.83e172T^{2} \) |
| 83 | \( 1 + 1.48e87T + 4.32e174T^{2} \) |
| 89 | \( 1 - 3.15e88T + 2.48e177T^{2} \) |
| 97 | \( 1 + 8.67e89T + 6.25e180T^{2} \) |
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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
−14.70434060788709801119681491921, −13.48851914736799849392136708130, −11.70279331720407519908165253359, −11.30431593572105218107547580339, −8.400976617496344736586585534097, −6.53526846695273951186242505097, −5.27211055226526977320336903362, −4.21902452762202469201490419899, −2.91870318452728912422447069082, −0.988309986932571347789043990027,
0.988309986932571347789043990027, 2.91870318452728912422447069082, 4.21902452762202469201490419899, 5.27211055226526977320336903362, 6.53526846695273951186242505097, 8.400976617496344736586585534097, 11.30431593572105218107547580339, 11.70279331720407519908165253359, 13.48851914736799849392136708130, 14.70434060788709801119681491921
