| L(s) = 1 | + 8·2-s − 21·3-s + 64·4-s − 551·5-s − 168·6-s + 62·7-s + 512·8-s − 1.74e3·9-s − 4.40e3·10-s − 1.33e3·11-s − 1.34e3·12-s + 1.50e3·13-s + 496·14-s + 1.15e4·15-s + 4.09e3·16-s − 2.99e4·17-s − 1.39e4·18-s + 2.95e4·19-s − 3.52e4·20-s − 1.30e3·21-s − 1.06e4·22-s + 3.14e4·23-s − 1.07e4·24-s + 2.25e5·25-s + 1.20e4·26-s + 8.25e4·27-s + 3.96e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.449·3-s + 1/2·4-s − 1.97·5-s − 0.317·6-s + 0.0683·7-s + 0.353·8-s − 0.798·9-s − 1.39·10-s − 0.301·11-s − 0.224·12-s + 0.189·13-s + 0.0483·14-s + 0.885·15-s + 1/4·16-s − 1.47·17-s − 0.564·18-s + 0.987·19-s − 0.985·20-s − 0.0306·21-s − 0.213·22-s + 0.539·23-s − 0.158·24-s + 2.88·25-s + 0.133·26-s + 0.807·27-s + 0.0341·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 11 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 + 7 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 551 T + p^{7} T^{2} \) |
| 7 | \( 1 - 62 T + p^{7} T^{2} \) |
| 13 | \( 1 - 1500 T + p^{7} T^{2} \) |
| 17 | \( 1 + 29930 T + p^{7} T^{2} \) |
| 19 | \( 1 - 29512 T + p^{7} T^{2} \) |
| 23 | \( 1 - 31499 T + p^{7} T^{2} \) |
| 29 | \( 1 + 2592 p T + p^{7} T^{2} \) |
| 31 | \( 1 + 235845 T + p^{7} T^{2} \) |
| 37 | \( 1 - 75507 T + p^{7} T^{2} \) |
| 41 | \( 1 + 270288 T + p^{7} T^{2} \) |
| 43 | \( 1 + 1028030 T + p^{7} T^{2} \) |
| 47 | \( 1 + 771840 T + p^{7} T^{2} \) |
| 53 | \( 1 - 765778 T + p^{7} T^{2} \) |
| 59 | \( 1 + 392007 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1248460 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3498133 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1101753 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1122996 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4362946 T + p^{7} T^{2} \) |
| 83 | \( 1 + 4437790 T + p^{7} T^{2} \) |
| 89 | \( 1 + 521233 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2129831 T + p^{7} T^{2} \) |
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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
−15.67784841495653445702087436245, −14.71915755285737184568829304128, −12.97247232877001412691906416846, −11.58278921408456870130900431424, −11.16147774391340200620550606267, −8.421704771950237687308760234901, −6.99953434010570401595223442140, −4.96251851453768340730172047267, −3.41343194002727054195300481298, 0,
3.41343194002727054195300481298, 4.96251851453768340730172047267, 6.99953434010570401595223442140, 8.421704771950237687308760234901, 11.16147774391340200620550606267, 11.58278921408456870130900431424, 12.97247232877001412691906416846, 14.71915755285737184568829304128, 15.67784841495653445702087436245
