| L(s) = 1 | + 2-s + 3-s − 4-s + 2·5-s + 6-s − 3·8-s + 9-s + 2·10-s − 12-s − 2·13-s + 2·15-s − 16-s − 6·17-s + 18-s + 4·19-s − 2·20-s − 4·23-s − 3·24-s − 25-s − 2·26-s + 27-s − 6·29-s + 2·30-s + 8·31-s + 5·32-s − 6·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s + 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.834·23-s − 0.612·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 1.11·29-s + 0.365·30-s + 1.43·31-s + 0.883·32-s − 1.02·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.609146459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609146459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.50115305627523489125953491503, −12.72324052115466884963991201077, −11.49684764509893737567106326450, −9.873976645501824723502592471470, −9.324362356599777608232228805650, −8.095096812813824172389012675232, −6.51036929960379395923767768849, −5.31003528872580210111956919467, −4.07864503838692042052899871035, −2.46985566708181710943572560797,
2.46985566708181710943572560797, 4.07864503838692042052899871035, 5.31003528872580210111956919467, 6.51036929960379395923767768849, 8.095096812813824172389012675232, 9.324362356599777608232228805650, 9.873976645501824723502592471470, 11.49684764509893737567106326450, 12.72324052115466884963991201077, 13.50115305627523489125953491503
