| L(s) = 1 | + 2-s − 3·3-s + 4-s + 5-s − 3·6-s + 7-s + 8-s + 6·9-s + 10-s − 2·11-s − 3·12-s + 7·13-s + 14-s − 3·15-s + 16-s + 4·17-s + 6·18-s − 19-s + 20-s − 3·21-s − 2·22-s − 4·23-s − 3·24-s + 25-s + 7·26-s − 9·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s − 0.866·12-s + 1.94·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s + 0.970·17-s + 1.41·18-s − 0.229·19-s + 0.223·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.612·24-s + 1/5·25-s + 1.37·26-s − 1.73·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 237910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 237910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.866991570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.866991570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 643 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| 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
−12.79676597036834, −12.48835121648270, −11.92787758189651, −11.48480749538538, −11.05312661816091, −10.87129833477653, −10.26672268110078, −9.944454432573276, −9.345230725175596, −8.568634076871074, −8.015118093485527, −7.679777625636882, −6.911644637817354, −6.414694967684334, −6.135538874233605, −5.599497221719567, −5.403103428723037, −4.791849025141621, −4.285985524367742, −3.648844081105417, −3.299673953832303, −2.171841891617845, −1.818814226360073, −0.9629472673699207, −0.6642970168984876,
0.6642970168984876, 0.9629472673699207, 1.818814226360073, 2.171841891617845, 3.299673953832303, 3.648844081105417, 4.285985524367742, 4.791849025141621, 5.403103428723037, 5.599497221719567, 6.135538874233605, 6.414694967684334, 6.911644637817354, 7.679777625636882, 8.015118093485527, 8.568634076871074, 9.345230725175596, 9.944454432573276, 10.26672268110078, 10.87129833477653, 11.05312661816091, 11.48480749538538, 11.92787758189651, 12.48835121648270, 12.79676597036834
