| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 7-s + 8-s − 2·9-s + 2·10-s − 5·11-s − 12-s − 14-s − 2·15-s + 16-s + 2·17-s − 2·18-s + 4·19-s + 2·20-s + 21-s − 5·22-s − 9·23-s − 24-s − 25-s + 5·27-s − 28-s − 2·30-s − 5·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.632·10-s − 1.50·11-s − 0.288·12-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s + 0.917·19-s + 0.447·20-s + 0.218·21-s − 1.06·22-s − 1.87·23-s − 0.204·24-s − 1/5·25-s + 0.962·27-s − 0.188·28-s − 0.365·30-s − 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.451475679654588564423601393948, −7.74916403073757463849691042581, −6.81102724841897540353349510344, −5.85647717055348692748649773331, −5.59205642825903658648469848604, −4.92801608368813666625605105316, −3.60688827299301057032966237980, −2.76325671186704681815328664832, −1.82291451221084311696910628828, 0,
1.82291451221084311696910628828, 2.76325671186704681815328664832, 3.60688827299301057032966237980, 4.92801608368813666625605105316, 5.59205642825903658648469848604, 5.85647717055348692748649773331, 6.81102724841897540353349510344, 7.74916403073757463849691042581, 8.451475679654588564423601393948
