| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s − 7-s + 3·8-s + 9-s + 10-s + 4·11-s + 12-s − 6·13-s + 14-s + 15-s − 16-s − 2·17-s − 18-s + 20-s + 21-s − 4·22-s − 23-s − 3·24-s + 25-s + 6·26-s − 27-s + 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.218·21-s − 0.852·22-s − 0.208·23-s − 0.612·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.689826211509155114573979136532, −7.81761317378765409612184255837, −7.11221641251139968421919176216, −6.46757385686276194163119881194, −5.34067514323999524648878050982, −4.49259103823795207312367096051, −3.96058678177192292607993886295, −2.52879661840008857202697720900, −1.11290859636845340340941393854, 0,
1.11290859636845340340941393854, 2.52879661840008857202697720900, 3.96058678177192292607993886295, 4.49259103823795207312367096051, 5.34067514323999524648878050982, 6.46757385686276194163119881194, 7.11221641251139968421919176216, 7.81761317378765409612184255837, 8.689826211509155114573979136532
