| L(s) = 1 | + 8·2-s + 64·4-s − 450·5-s − 568·7-s + 512·8-s − 3.60e3·10-s + 5.88e3·11-s + 2.85e3·13-s − 4.54e3·14-s + 4.09e3·16-s + 8.95e3·17-s + 6.85e3·19-s − 2.88e4·20-s + 4.70e4·22-s − 4.78e4·23-s + 1.24e5·25-s + 2.28e4·26-s − 3.63e4·28-s + 9.48e4·29-s − 2.64e4·31-s + 3.27e4·32-s + 7.16e4·34-s + 2.55e5·35-s + 9.32e4·37-s + 5.48e4·38-s − 2.30e5·40-s + 4.45e4·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.60·5-s − 0.625·7-s + 0.353·8-s − 1.13·10-s + 1.33·11-s + 0.360·13-s − 0.442·14-s + 1/4·16-s + 0.442·17-s + 0.229·19-s − 0.804·20-s + 0.941·22-s − 0.819·23-s + 1.59·25-s + 0.255·26-s − 0.312·28-s + 0.721·29-s − 0.159·31-s + 0.176·32-s + 0.312·34-s + 1.00·35-s + 0.302·37-s + 0.162·38-s − 0.569·40-s + 0.100·41-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 | \( 1 + 18 p^{2} T + p^{7} T^{2} \) |
| 7 | \( 1 + 568 T + p^{7} T^{2} \) |
| 11 | \( 1 - 5880 T + p^{7} T^{2} \) |
| 13 | \( 1 - 2858 T + p^{7} T^{2} \) |
| 17 | \( 1 - 8958 T + p^{7} T^{2} \) |
| 23 | \( 1 + 47832 T + p^{7} T^{2} \) |
| 29 | \( 1 - 94806 T + p^{7} T^{2} \) |
| 31 | \( 1 + 26428 T + p^{7} T^{2} \) |
| 37 | \( 1 - 93242 T + p^{7} T^{2} \) |
| 41 | \( 1 - 44514 T + p^{7} T^{2} \) |
| 43 | \( 1 + 21964 p T + p^{7} T^{2} \) |
| 47 | \( 1 - 713448 T + p^{7} T^{2} \) |
| 53 | \( 1 + 649218 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2059452 T + p^{7} T^{2} \) |
| 61 | \( 1 - 955574 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2926444 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2619840 T + p^{7} T^{2} \) |
| 73 | \( 1 + 6308278 T + p^{7} T^{2} \) |
| 79 | \( 1 + 7677100 T + p^{7} T^{2} \) |
| 83 | \( 1 - 413616 T + p^{7} T^{2} \) |
| 89 | \( 1 - 6215154 T + p^{7} T^{2} \) |
| 97 | \( 1 - 6963650 T + p^{7} T^{2} \) |
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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
−9.973069603179312634691884538467, −8.781093381709734581886853767809, −7.82744645013600638651062215564, −6.90510566849673053656321592593, −6.06329990234133656701756045360, −4.58487140768633184722277006305, −3.79764108749844718464079320061, −3.11625984687459788218352763616, −1.30842460008632747358902296295, 0,
1.30842460008632747358902296295, 3.11625984687459788218352763616, 3.79764108749844718464079320061, 4.58487140768633184722277006305, 6.06329990234133656701756045360, 6.90510566849673053656321592593, 7.82744645013600638651062215564, 8.781093381709734581886853767809, 9.973069603179312634691884538467
