| L(s) = 1 | − 4·2-s − 12.3·3-s + 16·4-s − 48.5·5-s + 49.5·6-s + 247.·7-s − 64·8-s − 89.5·9-s + 194.·10-s + 578.·11-s − 198.·12-s − 730.·13-s − 988.·14-s + 602.·15-s + 256·16-s − 661.·17-s + 358.·18-s + 1.18e3·19-s − 777.·20-s − 3.06e3·21-s − 2.31e3·22-s − 3.63e3·23-s + 792.·24-s − 763.·25-s + 2.92e3·26-s + 4.11e3·27-s + 3.95e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.794·3-s + 0.5·4-s − 0.869·5-s + 0.561·6-s + 1.90·7-s − 0.353·8-s − 0.368·9-s + 0.614·10-s + 1.44·11-s − 0.397·12-s − 1.19·13-s − 1.34·14-s + 0.690·15-s + 0.250·16-s − 0.554·17-s + 0.260·18-s + 0.754·19-s − 0.434·20-s − 1.51·21-s − 1.01·22-s − 1.43·23-s + 0.280·24-s − 0.244·25-s + 0.847·26-s + 1.08·27-s + 0.953·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 41 | \( 1 + 1.68e3T \) |
| good | 3 | \( 1 + 12.3T + 243T^{2} \) |
| 5 | \( 1 + 48.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 247.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 578.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 730.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 661.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.63e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.83e3T + 6.93e7T^{2} \) |
| 43 | \( 1 - 2.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 650.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.17e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.00e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.19e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.79e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.22e5T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.07596886870872444025349802736, −11.58323911563009847804476965868, −11.01429504683582316102844133347, −9.358594025407107207569799773785, −8.114521927470245081930576378181, −7.26037939943799699404819178995, −5.58490875887358646702318490558, −4.22368573812377351655829649360, −1.70885036230087623998601697121, 0,
1.70885036230087623998601697121, 4.22368573812377351655829649360, 5.58490875887358646702318490558, 7.26037939943799699404819178995, 8.114521927470245081930576378181, 9.358594025407107207569799773785, 11.01429504683582316102844133347, 11.58323911563009847804476965868, 12.07596886870872444025349802736
