| L(s) = 1 | − 8·2-s + 64·4-s + 47·5-s + 405·7-s − 512·8-s − 376·10-s + 5.78e3·11-s − 2.68e3·13-s − 3.24e3·14-s + 4.09e3·16-s − 2.21e4·17-s − 6.85e3·19-s + 3.00e3·20-s − 4.63e4·22-s − 1.27e4·23-s − 7.59e4·25-s + 2.14e4·26-s + 2.59e4·28-s + 2.07e5·29-s − 2.21e4·31-s − 3.27e4·32-s + 1.77e5·34-s + 1.90e4·35-s − 5.50e5·37-s + 5.48e4·38-s − 2.40e4·40-s + 2.06e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.168·5-s + 0.446·7-s − 0.353·8-s − 0.118·10-s + 1.31·11-s − 0.339·13-s − 0.315·14-s + 1/4·16-s − 1.09·17-s − 0.229·19-s + 0.0840·20-s − 0.927·22-s − 0.218·23-s − 0.971·25-s + 0.239·26-s + 0.223·28-s + 1.58·29-s − 0.133·31-s − 0.176·32-s + 0.773·34-s + 0.0750·35-s − 1.78·37-s + 0.162·38-s − 0.0594·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 - 47 T + p^{7} T^{2} \) |
| 7 | \( 1 - 405 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5789 T + p^{7} T^{2} \) |
| 13 | \( 1 + 2686 T + p^{7} T^{2} \) |
| 17 | \( 1 + 22167 T + p^{7} T^{2} \) |
| 23 | \( 1 + 12772 T + p^{7} T^{2} \) |
| 29 | \( 1 - 207538 T + p^{7} T^{2} \) |
| 31 | \( 1 + 22106 T + p^{7} T^{2} \) |
| 37 | \( 1 + 550160 T + p^{7} T^{2} \) |
| 41 | \( 1 - 206800 T + p^{7} T^{2} \) |
| 43 | \( 1 + 565547 T + p^{7} T^{2} \) |
| 47 | \( 1 + 176953 T + p^{7} T^{2} \) |
| 53 | \( 1 - 717230 T + p^{7} T^{2} \) |
| 59 | \( 1 + 193968 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2285819 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3373056 T + p^{7} T^{2} \) |
| 71 | \( 1 + 110068 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2640093 T + p^{7} T^{2} \) |
| 79 | \( 1 - 4870904 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5991996 T + p^{7} T^{2} \) |
| 89 | \( 1 + 3078666 T + p^{7} T^{2} \) |
| 97 | \( 1 - 682750 T + p^{7} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.782269388081837614097600097597, −8.907050024327330729330285013808, −8.188673196003073381458296256559, −6.96073187839780325109075442678, −6.30139805043244765172640164276, −4.90980494428152648662698756372, −3.74937523930197777364840500473, −2.25772793992696455758957733729, −1.33545459540286537341829805088, 0,
1.33545459540286537341829805088, 2.25772793992696455758957733729, 3.74937523930197777364840500473, 4.90980494428152648662698756372, 6.30139805043244765172640164276, 6.96073187839780325109075442678, 8.188673196003073381458296256559, 8.907050024327330729330285013808, 9.782269388081837614097600097597
