| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 2·11-s − 4·13-s + 14-s + 16-s + 2·17-s − 2·19-s − 2·22-s − 2·23-s − 4·26-s + 28-s − 29-s + 3·31-s + 32-s + 2·34-s + 3·37-s − 2·38-s − 3·41-s − 8·43-s − 2·44-s − 2·46-s − 3·47-s + 49-s − 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.603·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.458·19-s − 0.426·22-s − 0.417·23-s − 0.784·26-s + 0.188·28-s − 0.185·29-s + 0.538·31-s + 0.176·32-s + 0.342·34-s + 0.493·37-s − 0.324·38-s − 0.468·41-s − 1.21·43-s − 0.301·44-s − 0.294·46-s − 0.437·47-s + 1/7·49-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.20906285114713238848537743177, −6.73399222863261542765152012033, −5.79390251719677411967183062554, −5.26087626597508612615566793108, −4.62750622051817264182266069380, −3.95220276258807481447462122345, −2.97802737968854712175960737753, −2.37439655756410997287271375909, −1.44816100766767733490749194623, 0,
1.44816100766767733490749194623, 2.37439655756410997287271375909, 2.97802737968854712175960737753, 3.95220276258807481447462122345, 4.62750622051817264182266069380, 5.26087626597508612615566793108, 5.79390251719677411967183062554, 6.73399222863261542765152012033, 7.20906285114713238848537743177
