| L(s) = 1 | + 4·2-s + 15·3-s − 16·4-s + 60·6-s − 10·7-s − 192·8-s − 18·9-s − 121·11-s − 240·12-s + 1.14e3·13-s − 40·14-s − 256·16-s − 686·17-s − 72·18-s − 384·19-s − 150·21-s − 484·22-s − 3.70e3·23-s − 2.88e3·24-s + 4.59e3·26-s − 3.91e3·27-s + 160·28-s − 5.42e3·29-s − 6.44e3·31-s + 5.12e3·32-s − 1.81e3·33-s − 2.74e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.962·3-s − 1/2·4-s + 0.680·6-s − 0.0771·7-s − 1.06·8-s − 0.0740·9-s − 0.301·11-s − 0.481·12-s + 1.88·13-s − 0.0545·14-s − 1/4·16-s − 0.575·17-s − 0.0523·18-s − 0.244·19-s − 0.0742·21-s − 0.213·22-s − 1.46·23-s − 1.02·24-s + 1.33·26-s − 1.03·27-s + 0.0385·28-s − 1.19·29-s − 1.20·31-s + 0.883·32-s − 0.290·33-s − 0.407·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 3 | \( 1 - 5 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 10 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1148 T + p^{5} T^{2} \) |
| 17 | \( 1 + 686 T + p^{5} T^{2} \) |
| 19 | \( 1 + 384 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3709 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5424 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6443 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12063 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1528 T + p^{5} T^{2} \) |
| 43 | \( 1 - 4026 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7168 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29862 T + p^{5} T^{2} \) |
| 59 | \( 1 + 6461 T + p^{5} T^{2} \) |
| 61 | \( 1 + 16980 T + p^{5} T^{2} \) |
| 67 | \( 1 + 29999 T + p^{5} T^{2} \) |
| 71 | \( 1 - 31023 T + p^{5} T^{2} \) |
| 73 | \( 1 + 1924 T + p^{5} T^{2} \) |
| 79 | \( 1 - 65138 T + p^{5} T^{2} \) |
| 83 | \( 1 - 102714 T + p^{5} T^{2} \) |
| 89 | \( 1 - 17415 T + p^{5} T^{2} \) |
| 97 | \( 1 + 66905 T + p^{5} T^{2} \) |
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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
−10.61200942080076909054910336877, −9.278914622620082701260304109218, −8.713267165263330666469282906893, −7.898897459445359465829128190518, −6.31375396214525710120739091857, −5.43873736892736443824916066237, −3.94087720237654635375079762616, −3.42720455462506406132544889603, −1.95904305716157798578336938773, 0,
1.95904305716157798578336938773, 3.42720455462506406132544889603, 3.94087720237654635375079762616, 5.43873736892736443824916066237, 6.31375396214525710120739091857, 7.898897459445359465829128190518, 8.713267165263330666469282906893, 9.278914622620082701260304109218, 10.61200942080076909054910336877
