| L(s) = 1 | + 1.65·2-s − 125.·4-s + 348.·5-s + 493.·7-s − 418.·8-s + 575.·10-s − 3.98e3·11-s + 6.10e3·13-s + 814.·14-s + 1.53e4·16-s − 1.28e3·17-s − 1.87e3·19-s − 4.36e4·20-s − 6.59e3·22-s − 5.67e4·23-s + 4.30e4·25-s + 1.00e4·26-s − 6.17e4·28-s + 4.52e4·29-s − 1.42e5·31-s + 7.89e4·32-s − 2.12e3·34-s + 1.71e5·35-s − 1.22e5·37-s − 3.10e3·38-s − 1.45e5·40-s − 4.52e5·41-s + ⋯ |
| L(s) = 1 | + 0.146·2-s − 0.978·4-s + 1.24·5-s + 0.543·7-s − 0.288·8-s + 0.181·10-s − 0.903·11-s + 0.771·13-s + 0.0793·14-s + 0.936·16-s − 0.0634·17-s − 0.0627·19-s − 1.21·20-s − 0.131·22-s − 0.972·23-s + 0.551·25-s + 0.112·26-s − 0.531·28-s + 0.344·29-s − 0.859·31-s + 0.425·32-s − 0.00926·34-s + 0.676·35-s − 0.397·37-s − 0.00917·38-s − 0.359·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 1.65T + 128T^{2} \) |
| 5 | \( 1 - 348.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 493.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.98e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.10e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.28e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.87e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.67e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 4.52e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.42e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.22e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.52e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.37e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.56e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.39e6T + 1.17e12T^{2} \) |
| 61 | \( 1 + 1.59e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.26e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.28e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.88e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.18e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.07e6T + 8.07e13T^{2} \) |
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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
−9.239401003343436015494486943667, −8.520920521394565671393721863162, −7.63604867573717678430159257530, −6.17872211776770523217369192922, −5.53190815645911997854191656722, −4.73234101744915631343594623187, −3.59949216867429502122339941869, −2.29187091883316624098892649601, −1.29556202013056422471411262925, 0,
1.29556202013056422471411262925, 2.29187091883316624098892649601, 3.59949216867429502122339941869, 4.73234101744915631343594623187, 5.53190815645911997854191656722, 6.17872211776770523217369192922, 7.63604867573717678430159257530, 8.520920521394565671393721863162, 9.239401003343436015494486943667
