| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 2·13-s − 14-s + 16-s + 6·17-s − 4·19-s + 20-s + 23-s + 25-s − 2·26-s + 28-s − 6·29-s + 2·31-s − 32-s − 6·34-s + 35-s − 10·37-s + 4·38-s − 40-s + 6·41-s − 10·43-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 0.208·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s − 1.64·37-s + 0.648·38-s − 0.158·40-s + 0.937·41-s − 1.52·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.40702949393068, −16.05980876845937, −15.22919626167764, −14.71021377029966, −14.36557508057929, −13.47837319095917, −13.12773892107963, −12.25634423094448, −11.91150508662829, −11.11943520013016, −10.62719885034042, −10.13258730388683, −9.523346743586471, −8.877903806767429, −8.355452722516455, −7.788245522534528, −7.103282906638907, −6.478199092160003, −5.721065082793601, −5.301599853871394, −4.318914172339305, −3.497762063524084, −2.785612386319674, −1.759939112379866, −1.295593239012714, 0,
1.295593239012714, 1.759939112379866, 2.785612386319674, 3.497762063524084, 4.318914172339305, 5.301599853871394, 5.721065082793601, 6.478199092160003, 7.103282906638907, 7.788245522534528, 8.355452722516455, 8.877903806767429, 9.523346743586471, 10.13258730388683, 10.62719885034042, 11.11943520013016, 11.91150508662829, 12.25634423094448, 13.12773892107963, 13.47837319095917, 14.36557508057929, 14.71021377029966, 15.22919626167764, 16.05980876845937, 16.40702949393068
