| L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s − 2·11-s − 2·13-s + 2·14-s − 16-s + 2·17-s + 19-s − 2·22-s − 2·26-s − 2·28-s + 6·29-s − 4·31-s + 5·32-s + 2·34-s + 2·37-s + 38-s − 2·41-s − 10·43-s + 2·44-s − 3·49-s + 2·52-s − 10·53-s − 6·56-s + 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s − 0.603·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s + 0.485·17-s + 0.229·19-s − 0.426·22-s − 0.392·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.883·32-s + 0.342·34-s + 0.328·37-s + 0.162·38-s − 0.312·41-s − 1.52·43-s + 0.301·44-s − 3/7·49-s + 0.277·52-s − 1.37·53-s − 0.801·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.120537227606420514854208810374, −7.32810116920032613960134446669, −6.39528598506872809957504799300, −5.54866951843227001907889469845, −4.95037288511355902968279737092, −4.46836052875292129929427195599, −3.41937829594701693078815334604, −2.71414357576152678427321258899, −1.46933136897608423719987002566, 0,
1.46933136897608423719987002566, 2.71414357576152678427321258899, 3.41937829594701693078815334604, 4.46836052875292129929427195599, 4.95037288511355902968279737092, 5.54866951843227001907889469845, 6.39528598506872809957504799300, 7.32810116920032613960134446669, 8.120537227606420514854208810374
