| L(s) = 1 | − 216·2-s + 1.38e4·4-s − 5.21e4·5-s + 2.82e6·7-s + 4.07e6·8-s + 1.12e7·10-s − 2.05e7·11-s − 1.90e8·13-s − 6.09e8·14-s − 1.33e9·16-s − 1.64e9·17-s + 1.56e9·19-s − 7.23e8·20-s + 4.44e9·22-s − 9.45e9·23-s − 2.78e10·25-s + 4.10e10·26-s + 3.91e10·28-s + 3.69e10·29-s + 7.15e10·31-s + 1.54e11·32-s + 3.55e11·34-s − 1.47e11·35-s − 1.03e12·37-s − 3.37e11·38-s − 2.12e11·40-s − 1.64e12·41-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.423·4-s − 0.298·5-s + 1.29·7-s + 0.687·8-s + 0.355·10-s − 0.318·11-s − 0.840·13-s − 1.54·14-s − 1.24·16-s − 0.973·17-s + 0.401·19-s − 0.126·20-s + 0.380·22-s − 0.578·23-s − 0.911·25-s + 1.00·26-s + 0.549·28-s + 0.397·29-s + 0.467·31-s + 0.797·32-s + 1.16·34-s − 0.386·35-s − 1.79·37-s − 0.478·38-s − 0.205·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 27 p^{3} T + p^{15} T^{2} \) |
| 5 | \( 1 + 10422 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 403208 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 1871532 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 14621026 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 1646527986 T + p^{15} T^{2} \) |
| 19 | \( 1 - 1563257180 T + p^{15} T^{2} \) |
| 23 | \( 1 + 9451116072 T + p^{15} T^{2} \) |
| 29 | \( 1 - 36902568330 T + p^{15} T^{2} \) |
| 31 | \( 1 - 71588483552 T + p^{15} T^{2} \) |
| 37 | \( 1 + 1033652081554 T + p^{15} T^{2} \) |
| 41 | \( 1 + 1641974018202 T + p^{15} T^{2} \) |
| 43 | \( 1 + 492403109308 T + p^{15} T^{2} \) |
| 47 | \( 1 - 3410684952624 T + p^{15} T^{2} \) |
| 53 | \( 1 + 6797151655902 T + p^{15} T^{2} \) |
| 59 | \( 1 + 167099268060 p T + p^{15} T^{2} \) |
| 61 | \( 1 - 4931842626902 T + p^{15} T^{2} \) |
| 67 | \( 1 + 28837826625364 T + p^{15} T^{2} \) |
| 71 | \( 1 + 125050114914552 T + p^{15} T^{2} \) |
| 73 | \( 1 + 82171455513478 T + p^{15} T^{2} \) |
| 79 | \( 1 + 25413078694480 T + p^{15} T^{2} \) |
| 83 | \( 1 - 281736730890468 T + p^{15} T^{2} \) |
| 89 | \( 1 + 715618564776810 T + p^{15} T^{2} \) |
| 97 | \( 1 - 612786136081826 T + p^{15} T^{2} \) |
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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
−17.24494688754118335221713604009, −15.57654715717717754164700816674, −13.89436364238627509553579136078, −11.68843724269251637833842497358, −10.27947022692950441118883947846, −8.610199318624854444351906401992, −7.47352896676141408316324583551, −4.72245306563000751921607998772, −1.80981159385735569925893727447, 0,
1.80981159385735569925893727447, 4.72245306563000751921607998772, 7.47352896676141408316324583551, 8.610199318624854444351906401992, 10.27947022692950441118883947846, 11.68843724269251637833842497358, 13.89436364238627509553579136078, 15.57654715717717754164700816674, 17.24494688754118335221713604009
