| L(s) = 1 | − 1.55e16·2-s + 1.35e25·3-s + 7.99e31·4-s − 3.05e37·5-s − 2.10e41·6-s + 6.30e44·7-s + 1.28e48·8-s − 9.43e50·9-s + 4.75e53·10-s − 3.14e55·11-s + 1.08e57·12-s − 3.79e59·13-s − 9.80e60·14-s − 4.13e62·15-s − 3.29e64·16-s + 8.88e64·17-s + 1.46e67·18-s + 2.23e68·19-s − 2.44e69·20-s + 8.53e69·21-s + 4.88e71·22-s − 6.00e72·23-s + 1.73e73·24-s + 3.15e74·25-s + 5.90e75·26-s − 2.80e76·27-s + 5.03e76·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.403·3-s + 0.492·4-s − 1.22·5-s − 0.492·6-s + 0.386·7-s + 0.619·8-s − 0.837·9-s + 1.50·10-s − 0.606·11-s + 0.198·12-s − 0.962·13-s − 0.471·14-s − 0.495·15-s − 1.24·16-s + 0.131·17-s + 1.02·18-s + 0.863·19-s − 0.605·20-s + 0.155·21-s + 0.740·22-s − 0.843·23-s + 0.249·24-s + 0.512·25-s + 1.17·26-s − 0.740·27-s + 0.190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(108-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+53.5) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(54)\) |
\(\approx\) |
\(0.2066061565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2066061565\) |
| \(L(\frac{109}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.55e16T + 1.62e32T^{2} \) |
| 3 | \( 1 - 1.35e25T + 1.12e51T^{2} \) |
| 5 | \( 1 + 3.05e37T + 6.16e74T^{2} \) |
| 7 | \( 1 - 6.30e44T + 2.66e90T^{2} \) |
| 11 | \( 1 + 3.14e55T + 2.68e111T^{2} \) |
| 13 | \( 1 + 3.79e59T + 1.55e119T^{2} \) |
| 17 | \( 1 - 8.88e64T + 4.55e131T^{2} \) |
| 19 | \( 1 - 2.23e68T + 6.70e136T^{2} \) |
| 23 | \( 1 + 6.00e72T + 5.06e145T^{2} \) |
| 29 | \( 1 + 1.13e77T + 2.99e156T^{2} \) |
| 31 | \( 1 + 1.07e80T + 3.76e159T^{2} \) |
| 37 | \( 1 + 1.32e84T + 6.27e167T^{2} \) |
| 41 | \( 1 + 2.90e86T + 3.69e172T^{2} \) |
| 43 | \( 1 - 2.19e87T + 6.04e174T^{2} \) |
| 47 | \( 1 - 2.36e89T + 8.21e178T^{2} \) |
| 53 | \( 1 + 8.73e91T + 3.14e184T^{2} \) |
| 59 | \( 1 + 7.08e94T + 3.02e189T^{2} \) |
| 61 | \( 1 + 4.17e95T + 1.07e191T^{2} \) |
| 67 | \( 1 + 5.09e97T + 2.45e195T^{2} \) |
| 71 | \( 1 - 1.86e99T + 1.21e198T^{2} \) |
| 73 | \( 1 - 5.87e99T + 2.37e199T^{2} \) |
| 79 | \( 1 - 5.08e101T + 1.11e203T^{2} \) |
| 83 | \( 1 + 2.20e102T + 2.19e205T^{2} \) |
| 89 | \( 1 - 1.22e104T + 3.84e208T^{2} \) |
| 97 | \( 1 - 1.43e106T + 3.84e212T^{2} \) |
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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
−14.08486073106460431969560560319, −12.00720438280682192217818197621, −10.75379444413961715268904542001, −9.241576746687495431868998325074, −8.044679426408810224684445543003, −7.46075582095281438228791918153, −5.04519656184348119532911071202, −3.46936195921916403428326429912, −1.93969587283282521293714230788, −0.26730979185041636452216199213,
0.26730979185041636452216199213, 1.93969587283282521293714230788, 3.46936195921916403428326429912, 5.04519656184348119532911071202, 7.46075582095281438228791918153, 8.044679426408810224684445543003, 9.241576746687495431868998325074, 10.75379444413961715268904542001, 12.00720438280682192217818197621, 14.08486073106460431969560560319
