| L(s) = 1 | + 38.1·2-s − 243·3-s − 590.·4-s + 118.·5-s − 9.27e3·6-s − 2.49e4·7-s − 1.00e5·8-s + 5.90e4·9-s + 4.51e3·10-s + 7.83e4·11-s + 1.43e5·12-s + 5.16e5·13-s − 9.54e5·14-s − 2.87e4·15-s − 2.63e6·16-s + 7.07e5·17-s + 2.25e6·18-s + 1.93e7·19-s − 6.98e4·20-s + 6.07e6·21-s + 2.98e6·22-s − 4.56e6·23-s + 2.44e7·24-s − 4.88e7·25-s + 1.97e7·26-s − 1.43e7·27-s + 1.47e7·28-s + ⋯ |
| L(s) = 1 | + 0.843·2-s − 0.577·3-s − 0.288·4-s + 0.0169·5-s − 0.487·6-s − 0.562·7-s − 1.08·8-s + 0.333·9-s + 0.0142·10-s + 0.146·11-s + 0.166·12-s + 0.385·13-s − 0.474·14-s − 0.00978·15-s − 0.628·16-s + 0.120·17-s + 0.281·18-s + 1.79·19-s − 0.00488·20-s + 0.324·21-s + 0.123·22-s − 0.147·23-s + 0.627·24-s − 0.999·25-s + 0.325·26-s − 0.192·27-s + 0.162·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 - 38.1T + 2.04e3T^{2} \) |
| 5 | \( 1 - 118.T + 4.88e7T^{2} \) |
| 7 | \( 1 + 2.49e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 7.83e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 5.16e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 7.07e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.93e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 4.56e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.21e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 5.43e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.64e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.48e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.12e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.05e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.10e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 9.70e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.88e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.71e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 6.45e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.94e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.19e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.13e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.49e10T + 7.15e21T^{2} \) |
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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
−10.08486717665929320530251052265, −9.394326353231305728436390247129, −8.083601275578034508340096954703, −6.69958477382988573700915643259, −5.82502906543812463566420440033, −4.96002870120936324723807732712, −3.85032285987182164901737666583, −2.92742522070789488015695941352, −1.14596099144472147764748238629, 0,
1.14596099144472147764748238629, 2.92742522070789488015695941352, 3.85032285987182164901737666583, 4.96002870120936324723807732712, 5.82502906543812463566420440033, 6.69958477382988573700915643259, 8.083601275578034508340096954703, 9.394326353231305728436390247129, 10.08486717665929320530251052265
