| L(s) = 1 | − 32·2-s + 1.02e3·4-s − 2.58e3·5-s + 1.68e4·7-s − 3.27e4·8-s + 8.28e4·10-s − 7.32e5·11-s + 2.13e6·13-s − 5.37e5·14-s + 1.04e6·16-s + 1.05e6·17-s − 8.10e6·19-s − 2.64e6·20-s + 2.34e7·22-s + 2.20e7·23-s − 4.21e7·25-s − 6.84e7·26-s + 1.72e7·28-s − 7.36e7·29-s + 1.29e8·31-s − 3.35e7·32-s − 3.36e7·34-s − 4.34e7·35-s + 3.62e8·37-s + 2.59e8·38-s + 8.47e7·40-s + 5.37e8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.370·5-s + 0.377·7-s − 0.353·8-s + 0.261·10-s − 1.37·11-s + 1.59·13-s − 0.267·14-s + 0.250·16-s + 0.179·17-s − 0.751·19-s − 0.185·20-s + 0.970·22-s + 0.715·23-s − 0.862·25-s − 1.13·26-s + 0.188·28-s − 0.666·29-s + 0.814·31-s − 0.176·32-s − 0.127·34-s − 0.139·35-s + 0.859·37-s + 0.531·38-s + 0.130·40-s + 0.724·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 32T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 1.68e4T \) |
| good | 5 | \( 1 + 2.58e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 7.32e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.13e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.05e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.10e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.20e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 7.36e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.29e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.62e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.37e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 3.82e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.47e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.50e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 5.98e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 3.71e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 2.03e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.37e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.08e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.32e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.56e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 7.17e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.89e10T + 7.15e21T^{2} \) |
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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
−10.85204421592038676891895115905, −9.695907853919273224244779758735, −8.369439576113150356606595500305, −7.919495309519550128113795967317, −6.53878275885026191810083165840, −5.35205818298584841772683363267, −3.86174109825585214260971500890, −2.52072409613933525505780149112, −1.20928172919680905655816542643, 0,
1.20928172919680905655816542643, 2.52072409613933525505780149112, 3.86174109825585214260971500890, 5.35205818298584841772683363267, 6.53878275885026191810083165840, 7.919495309519550128113795967317, 8.369439576113150356606595500305, 9.695907853919273224244779758735, 10.85204421592038676891895115905
