| L(s) = 1 | + 8·2-s + 64·4-s + 297·5-s + 343·7-s + 512·8-s + 2.37e3·10-s − 378·11-s − 4.54e3·13-s + 2.74e3·14-s + 4.09e3·16-s − 2.16e4·17-s − 4.33e4·19-s + 1.90e4·20-s − 3.02e3·22-s − 8.60e4·23-s + 1.00e4·25-s − 3.63e4·26-s + 2.19e4·28-s − 2.15e4·29-s − 2.98e5·31-s + 3.27e4·32-s − 1.72e5·34-s + 1.01e5·35-s + 4.52e5·37-s − 3.46e5·38-s + 1.52e5·40-s − 8.03e5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.06·5-s + 0.377·7-s + 0.353·8-s + 0.751·10-s − 0.0856·11-s − 0.573·13-s + 0.267·14-s + 1/4·16-s − 1.06·17-s − 1.44·19-s + 0.531·20-s − 0.0605·22-s − 1.47·23-s + 0.129·25-s − 0.405·26-s + 0.188·28-s − 0.164·29-s − 1.80·31-s + 0.176·32-s − 0.754·34-s + 0.401·35-s + 1.46·37-s − 1.02·38-s + 0.375·40-s − 1.81·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 297 T + p^{7} T^{2} \) |
| 11 | \( 1 + 378 T + p^{7} T^{2} \) |
| 13 | \( 1 + 4540 T + p^{7} T^{2} \) |
| 17 | \( 1 + 21603 T + p^{7} T^{2} \) |
| 19 | \( 1 + 43306 T + p^{7} T^{2} \) |
| 23 | \( 1 + 86094 T + p^{7} T^{2} \) |
| 29 | \( 1 + 21570 T + p^{7} T^{2} \) |
| 31 | \( 1 + 298948 T + p^{7} T^{2} \) |
| 37 | \( 1 - 452117 T + p^{7} T^{2} \) |
| 41 | \( 1 + 803109 T + p^{7} T^{2} \) |
| 43 | \( 1 - 201293 T + p^{7} T^{2} \) |
| 47 | \( 1 - 411081 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1283826 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2628021 T + p^{7} T^{2} \) |
| 61 | \( 1 + 3258874 T + p^{7} T^{2} \) |
| 67 | \( 1 - 4158788 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1889280 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2209466 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2478655 T + p^{7} T^{2} \) |
| 83 | \( 1 + 472785 T + p^{7} T^{2} \) |
| 89 | \( 1 - 9461514 T + p^{7} T^{2} \) |
| 97 | \( 1 - 13756166 T + p^{7} T^{2} \) |
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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
−9.854102874702565050105091843041, −8.851419827703563091170584761015, −7.73514956210706780875900636549, −6.57048940596650002650638181979, −5.86060747722703455208243877624, −4.85700894549125805347499807111, −3.89731877163442129367847954488, −2.30281896115281793599682308580, −1.85859003870560925859261492649, 0,
1.85859003870560925859261492649, 2.30281896115281793599682308580, 3.89731877163442129367847954488, 4.85700894549125805347499807111, 5.86060747722703455208243877624, 6.57048940596650002650638181979, 7.73514956210706780875900636549, 8.851419827703563091170584761015, 9.854102874702565050105091843041
