| L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 47·5-s − 216·6-s + 405·7-s + 512·8-s + 729·9-s − 376·10-s − 5.78e3·11-s − 1.72e3·12-s − 2.68e3·13-s + 3.24e3·14-s + 1.26e3·15-s + 4.09e3·16-s + 2.21e4·17-s + 5.83e3·18-s − 6.85e3·19-s − 3.00e3·20-s − 1.09e4·21-s − 4.63e4·22-s + 1.27e4·23-s − 1.38e4·24-s − 7.59e4·25-s − 2.14e4·26-s − 1.96e4·27-s + 2.59e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.168·5-s − 0.408·6-s + 0.446·7-s + 0.353·8-s + 1/3·9-s − 0.118·10-s − 1.31·11-s − 0.288·12-s − 0.339·13-s + 0.315·14-s + 0.0970·15-s + 1/4·16-s + 1.09·17-s + 0.235·18-s − 0.229·19-s − 0.0840·20-s − 0.257·21-s − 0.927·22-s + 0.218·23-s − 0.204·24-s − 0.971·25-s − 0.239·26-s − 0.192·27-s + 0.223·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{3} T \) |
| 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 | \( 1 - 682750 T + p^{7} T^{2} \) |
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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
−11.79720675325463523426742519760, −10.84923871288412345146761295316, −9.901588069461961382290238424363, −8.100909929829619758676343472915, −7.18308674433257004634514228443, −5.66849599231860896786632213137, −4.96040819257631042224521877633, −3.46993606252197749613736715241, −1.85789408211078863752998635578, 0,
1.85789408211078863752998635578, 3.46993606252197749613736715241, 4.96040819257631042224521877633, 5.66849599231860896786632213137, 7.18308674433257004634514228443, 8.100909929829619758676343472915, 9.901588069461961382290238424363, 10.84923871288412345146761295316, 11.79720675325463523426742519760
