| L(s) = 1 | + 5.06e9·2-s − 7.82e15·3-s − 1.21e20·4-s − 8.33e22·5-s − 3.96e25·6-s − 1.15e28·7-s − 1.36e30·8-s − 3.14e31·9-s − 4.22e32·10-s − 1.09e35·11-s + 9.53e35·12-s + 1.54e37·13-s − 5.83e37·14-s + 6.52e38·15-s + 1.10e40·16-s + 8.24e40·17-s − 1.59e41·18-s − 5.93e42·19-s + 1.01e43·20-s + 9.00e43·21-s − 5.53e44·22-s + 2.28e45·23-s + 1.06e46·24-s − 6.08e46·25-s + 7.81e46·26-s + 9.71e47·27-s + 1.40e48·28-s + ⋯ |
| L(s) = 1 | + 0.417·2-s − 0.812·3-s − 0.826·4-s − 0.320·5-s − 0.338·6-s − 0.562·7-s − 0.761·8-s − 0.339·9-s − 0.133·10-s − 1.41·11-s + 0.671·12-s + 0.743·13-s − 0.234·14-s + 0.260·15-s + 0.508·16-s + 0.496·17-s − 0.141·18-s − 0.861·19-s + 0.264·20-s + 0.457·21-s − 0.591·22-s + 0.550·23-s + 0.618·24-s − 0.897·25-s + 0.310·26-s + 1.08·27-s + 0.464·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(68-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+67/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(34)\) |
\(\approx\) |
\(0.5873099641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5873099641\) |
| \(L(\frac{69}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 5.06e9T + 1.47e20T^{2} \) |
| 3 | \( 1 + 7.82e15T + 9.27e31T^{2} \) |
| 5 | \( 1 + 8.33e22T + 6.77e46T^{2} \) |
| 7 | \( 1 + 1.15e28T + 4.18e56T^{2} \) |
| 11 | \( 1 + 1.09e35T + 5.93e69T^{2} \) |
| 13 | \( 1 - 1.54e37T + 4.30e74T^{2} \) |
| 17 | \( 1 - 8.24e40T + 2.75e82T^{2} \) |
| 19 | \( 1 + 5.93e42T + 4.74e85T^{2} \) |
| 23 | \( 1 - 2.28e45T + 1.72e91T^{2} \) |
| 29 | \( 1 - 1.07e49T + 9.56e97T^{2} \) |
| 31 | \( 1 - 1.32e50T + 8.34e99T^{2} \) |
| 37 | \( 1 + 5.20e52T + 1.17e105T^{2} \) |
| 41 | \( 1 + 1.63e54T + 1.13e108T^{2} \) |
| 43 | \( 1 - 5.64e54T + 2.76e109T^{2} \) |
| 47 | \( 1 - 2.41e55T + 1.07e112T^{2} \) |
| 53 | \( 1 - 2.84e57T + 3.36e115T^{2} \) |
| 59 | \( 1 + 3.30e59T + 4.43e118T^{2} \) |
| 61 | \( 1 + 6.89e59T + 4.14e119T^{2} \) |
| 67 | \( 1 - 2.58e61T + 2.22e122T^{2} \) |
| 71 | \( 1 + 1.34e62T + 1.08e124T^{2} \) |
| 73 | \( 1 + 1.11e62T + 6.96e124T^{2} \) |
| 79 | \( 1 - 1.23e63T + 1.38e127T^{2} \) |
| 83 | \( 1 - 3.41e63T + 3.78e128T^{2} \) |
| 89 | \( 1 - 3.04e65T + 4.06e130T^{2} \) |
| 97 | \( 1 - 2.29e66T + 1.29e133T^{2} \) |
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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
−17.43012023292266600367403828535, −15.65853967129481623159398249662, −13.60982873229299309727506926887, −12.22874343166330535552594730334, −10.41186364793203780853614668045, −8.398297559593765716015857212247, −6.09078674297176093913354331349, −4.85449009180606295178890287890, −3.15911341946688949251242986902, −0.46173712903443472817543896296,
0.46173712903443472817543896296, 3.15911341946688949251242986902, 4.85449009180606295178890287890, 6.09078674297176093913354331349, 8.398297559593765716015857212247, 10.41186364793203780853614668045, 12.22874343166330535552594730334, 13.60982873229299309727506926887, 15.65853967129481623159398249662, 17.43012023292266600367403828535
