| L(s) = 1 | + 11.7·2-s + 9.96·4-s + 389.·5-s − 1.24e3·7-s − 1.38e3·8-s + 4.57e3·10-s − 1.33e3·11-s − 3.84e3·13-s − 1.46e4·14-s − 1.75e4·16-s − 2.41e4·17-s + 5.45e3·19-s + 3.88e3·20-s − 1.56e4·22-s + 6.39e4·23-s + 7.39e4·25-s − 4.51e4·26-s − 1.24e4·28-s − 1.78e5·29-s − 1.85e5·31-s − 2.88e4·32-s − 2.83e5·34-s − 4.87e5·35-s − 4.09e5·37-s + 6.41e4·38-s − 5.40e5·40-s + 6.75e5·41-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 0.0778·4-s + 1.39·5-s − 1.37·7-s − 0.957·8-s + 1.44·10-s − 0.301·11-s − 0.484·13-s − 1.42·14-s − 1.07·16-s − 1.19·17-s + 0.182·19-s + 0.108·20-s − 0.313·22-s + 1.09·23-s + 0.946·25-s − 0.503·26-s − 0.107·28-s − 1.35·29-s − 1.11·31-s − 0.155·32-s − 1.23·34-s − 1.92·35-s − 1.33·37-s + 0.189·38-s − 1.33·40-s + 1.53·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + 1.33e3T \) |
| good | 2 | \( 1 - 11.7T + 128T^{2} \) |
| 5 | \( 1 - 389.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.24e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 3.84e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.41e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.45e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.78e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.85e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.75e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.89e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.94e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.78e4T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.78e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.95e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.08e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.95e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.08e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.14e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.30e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.38e6T + 8.07e13T^{2} \) |
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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
−12.63350451708062607536969416832, −10.94960940760774202898195114265, −9.576649895932213845509059972004, −9.148202811016201891654435151500, −6.88918192191581221327913991406, −5.97237890600531487472401183971, −5.03118646393612900016473843743, −3.43852362472737562098772401008, −2.27040256048164475300617215260, 0,
2.27040256048164475300617215260, 3.43852362472737562098772401008, 5.03118646393612900016473843743, 5.97237890600531487472401183971, 6.88918192191581221327913991406, 9.148202811016201891654435151500, 9.576649895932213845509059972004, 10.94960940760774202898195114265, 12.63350451708062607536969416832
