| L(s) = 1 | − 2.78e8·2-s − 3.57e13·3-s − 6.67e16·4-s + 2.54e19·5-s + 9.94e21·6-s + 4.82e23·7-s + 5.86e25·8-s − 2.90e26·9-s − 7.06e27·10-s + 7.50e29·11-s + 2.38e30·12-s − 7.80e30·13-s − 1.34e32·14-s − 9.08e32·15-s − 6.69e33·16-s + 8.46e34·17-s + 8.08e34·18-s − 4.77e36·19-s − 1.69e36·20-s − 1.72e37·21-s − 2.08e38·22-s − 8.01e38·23-s − 2.09e39·24-s − 6.29e39·25-s + 2.16e39·26-s + 6.65e40·27-s − 3.22e40·28-s + ⋯ |
| L(s) = 1 | − 0.732·2-s − 0.902·3-s − 0.463·4-s + 0.304·5-s + 0.661·6-s + 0.396·7-s + 1.07·8-s − 0.185·9-s − 0.223·10-s + 1.56·11-s + 0.418·12-s − 0.139·13-s − 0.290·14-s − 0.275·15-s − 0.322·16-s + 0.724·17-s + 0.135·18-s − 1.71·19-s − 0.141·20-s − 0.358·21-s − 1.15·22-s − 1.24·23-s − 0.967·24-s − 0.907·25-s + 0.102·26-s + 1.06·27-s − 0.183·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(58-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+57/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(29)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{59}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.78e8T + 1.44e17T^{2} \) |
| 3 | \( 1 + 3.57e13T + 1.57e27T^{2} \) |
| 5 | \( 1 - 2.54e19T + 6.93e39T^{2} \) |
| 7 | \( 1 - 4.82e23T + 1.48e48T^{2} \) |
| 11 | \( 1 - 7.50e29T + 2.28e59T^{2} \) |
| 13 | \( 1 + 7.80e30T + 3.12e63T^{2} \) |
| 17 | \( 1 - 8.46e34T + 1.36e70T^{2} \) |
| 19 | \( 1 + 4.77e36T + 7.74e72T^{2} \) |
| 23 | \( 1 + 8.01e38T + 4.15e77T^{2} \) |
| 29 | \( 1 - 4.53e41T + 2.27e83T^{2} \) |
| 31 | \( 1 - 4.59e42T + 1.01e85T^{2} \) |
| 37 | \( 1 + 9.56e43T + 2.44e89T^{2} \) |
| 41 | \( 1 + 2.44e45T + 8.48e91T^{2} \) |
| 43 | \( 1 + 4.83e46T + 1.28e93T^{2} \) |
| 47 | \( 1 - 4.58e47T + 2.03e95T^{2} \) |
| 53 | \( 1 + 9.07e48T + 1.92e98T^{2} \) |
| 59 | \( 1 - 7.50e49T + 8.68e100T^{2} \) |
| 61 | \( 1 + 7.06e50T + 5.80e101T^{2} \) |
| 67 | \( 1 + 5.94e51T + 1.21e104T^{2} \) |
| 71 | \( 1 + 5.65e52T + 3.32e105T^{2} \) |
| 73 | \( 1 + 1.37e53T + 1.61e106T^{2} \) |
| 79 | \( 1 + 5.55e53T + 1.46e108T^{2} \) |
| 83 | \( 1 + 3.41e54T + 2.44e109T^{2} \) |
| 89 | \( 1 + 3.57e55T + 1.30e111T^{2} \) |
| 97 | \( 1 - 6.10e56T + 1.76e113T^{2} \) |
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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
−17.62728381747005162262396267369, −16.89905927196541329743604102547, −14.16664450784940148135281846966, −11.88546095742190219724901706354, −10.14981542898063551358400840658, −8.463727629061523807859670302970, −6.22178203762451426384050184986, −4.40982350389365240746261505984, −1.44458006649267949059849455512, 0,
1.44458006649267949059849455512, 4.40982350389365240746261505984, 6.22178203762451426384050184986, 8.463727629061523807859670302970, 10.14981542898063551358400840658, 11.88546095742190219724901706354, 14.16664450784940148135281846966, 16.89905927196541329743604102547, 17.62728381747005162262396267369
