| L(s) = 1 | + 85.0·2-s + 468.·3-s + 5.18e3·4-s − 69.9·5-s + 3.98e4·6-s − 8.00e4·7-s + 2.66e5·8-s + 4.23e4·9-s − 5.94e3·10-s − 3.79e5·11-s + 2.42e6·12-s − 3.71e5·13-s − 6.80e6·14-s − 3.27e4·15-s + 1.20e7·16-s + 4.77e6·17-s + 3.59e6·18-s + 1.86e7·19-s − 3.62e5·20-s − 3.74e7·21-s − 3.22e7·22-s + 8.06e5·23-s + 1.25e8·24-s − 4.88e7·25-s − 3.15e7·26-s − 6.31e7·27-s − 4.15e8·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 1.11·3-s + 2.53·4-s − 0.0100·5-s + 2.09·6-s − 1.79·7-s + 2.88·8-s + 0.238·9-s − 0.0188·10-s − 0.710·11-s + 2.81·12-s − 0.277·13-s − 3.38·14-s − 0.0111·15-s + 2.88·16-s + 0.816·17-s + 0.448·18-s + 1.72·19-s − 0.0253·20-s − 2.00·21-s − 1.33·22-s + 0.0261·23-s + 3.20·24-s − 0.999·25-s − 0.521·26-s − 0.847·27-s − 4.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(5.852911170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.852911170\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + 3.71e5T \) |
| good | 2 | \( 1 - 85.0T + 2.04e3T^{2} \) |
| 3 | \( 1 - 468.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 69.9T + 4.88e7T^{2} \) |
| 7 | \( 1 + 8.00e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 3.79e5T + 2.85e11T^{2} \) |
| 17 | \( 1 - 4.77e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.86e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 8.06e5T + 9.52e14T^{2} \) |
| 29 | \( 1 - 2.93e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.00e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.69e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 3.21e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.46e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.51e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.37e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 6.75e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 3.87e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 4.04e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.92e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.55e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.57e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.99e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.57e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.22e9T + 7.15e21T^{2} \) |
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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
−16.23625636956089159872063262723, −15.42590105250487361591386386019, −14.02135393429368926325523429558, −13.28368993255537165736902762083, −12.08629749749247199385420184668, −9.882773425873078138308082365197, −7.38148207344697534159889621095, −5.70692251251252743264942329595, −3.51948179168283958510337182268, −2.73649783983132868893057936711,
2.73649783983132868893057936711, 3.51948179168283958510337182268, 5.70692251251252743264942329595, 7.38148207344697534159889621095, 9.882773425873078138308082365197, 12.08629749749247199385420184668, 13.28368993255537165736902762083, 14.02135393429368926325523429558, 15.42590105250487361591386386019, 16.23625636956089159872063262723
