| L(s) = 1 | − 8·2-s + 64·4-s + 105·5-s − 937·7-s − 512·8-s − 840·10-s + 5.94e3·11-s + 68·13-s + 7.49e3·14-s + 4.09e3·16-s − 5.40e3·17-s − 4.83e4·19-s + 6.72e3·20-s − 4.75e4·22-s − 642·23-s − 6.71e4·25-s − 544·26-s − 5.99e4·28-s − 1.25e5·29-s − 1.61e5·31-s − 3.27e4·32-s + 4.32e4·34-s − 9.83e4·35-s − 4.14e5·37-s + 3.87e5·38-s − 5.37e4·40-s − 6.27e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.375·5-s − 1.03·7-s − 0.353·8-s − 0.265·10-s + 1.34·11-s + 0.00858·13-s + 0.730·14-s + 1/4·16-s − 0.266·17-s − 1.61·19-s + 0.187·20-s − 0.951·22-s − 0.0110·23-s − 0.858·25-s − 0.00607·26-s − 0.516·28-s − 0.958·29-s − 0.972·31-s − 0.176·32-s + 0.188·34-s − 0.387·35-s − 1.34·37-s + 1.14·38-s − 0.132·40-s − 1.42·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 5 | \( 1 - 21 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 937 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5943 T + p^{7} T^{2} \) |
| 13 | \( 1 - 68 T + p^{7} T^{2} \) |
| 17 | \( 1 + 5400 T + p^{7} T^{2} \) |
| 19 | \( 1 + 48382 T + p^{7} T^{2} \) |
| 23 | \( 1 + 642 T + p^{7} T^{2} \) |
| 29 | \( 1 + 125934 T + p^{7} T^{2} \) |
| 31 | \( 1 + 161275 T + p^{7} T^{2} \) |
| 37 | \( 1 + 414286 T + p^{7} T^{2} \) |
| 41 | \( 1 + 627474 T + p^{7} T^{2} \) |
| 43 | \( 1 - 570590 T + p^{7} T^{2} \) |
| 47 | \( 1 - 538698 T + p^{7} T^{2} \) |
| 53 | \( 1 - 356283 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2910828 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2684168 T + p^{7} T^{2} \) |
| 67 | \( 1 - 2681078 T + p^{7} T^{2} \) |
| 71 | \( 1 + 3705480 T + p^{7} T^{2} \) |
| 73 | \( 1 + 153151 T + p^{7} T^{2} \) |
| 79 | \( 1 + 7579288 T + p^{7} T^{2} \) |
| 83 | \( 1 - 9345999 T + p^{7} T^{2} \) |
| 89 | \( 1 - 4033602 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5754097 T + p^{7} T^{2} \) |
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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
−13.19938004272634443308698646907, −12.08271888829144173923600617611, −10.74355637572772035475981799907, −9.579173007622124307402545033287, −8.755194742007829601391740687267, −6.99217514962894239360359026809, −6.04199280540018811141449333702, −3.75846308698384581055486347518, −1.89004323774433251467721670865, 0,
1.89004323774433251467721670865, 3.75846308698384581055486347518, 6.04199280540018811141449333702, 6.99217514962894239360359026809, 8.755194742007829601391740687267, 9.579173007622124307402545033287, 10.74355637572772035475981799907, 12.08271888829144173923600617611, 13.19938004272634443308698646907
