| L(s) = 1 | + 8·2-s + 12·3-s + 64·4-s + 210·5-s + 96·6-s − 1.01e3·7-s + 512·8-s − 2.04e3·9-s + 1.68e3·10-s − 1.09e3·11-s + 768·12-s − 8.12e3·14-s + 2.52e3·15-s + 4.09e3·16-s + 1.47e4·17-s − 1.63e4·18-s + 3.99e4·19-s + 1.34e4·20-s − 1.21e4·21-s − 8.73e3·22-s + 6.87e4·23-s + 6.14e3·24-s − 3.40e4·25-s − 5.07e4·27-s − 6.50e4·28-s − 1.02e5·29-s + 2.01e4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.256·3-s + 1/2·4-s + 0.751·5-s + 0.181·6-s − 1.11·7-s + 0.353·8-s − 0.934·9-s + 0.531·10-s − 0.247·11-s + 0.128·12-s − 0.791·14-s + 0.192·15-s + 1/4·16-s + 0.725·17-s − 0.660·18-s + 1.33·19-s + 0.375·20-s − 0.287·21-s − 0.174·22-s + 1.17·23-s + 0.0907·24-s − 0.435·25-s − 0.496·27-s − 0.559·28-s − 0.780·29-s + 0.136·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{3} T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 4 p T + p^{7} T^{2} \) |
| 5 | \( 1 - 42 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1016 T + p^{7} T^{2} \) |
| 11 | \( 1 + 1092 T + p^{7} T^{2} \) |
| 17 | \( 1 - 14706 T + p^{7} T^{2} \) |
| 19 | \( 1 - 39940 T + p^{7} T^{2} \) |
| 23 | \( 1 - 68712 T + p^{7} T^{2} \) |
| 29 | \( 1 + 102570 T + p^{7} T^{2} \) |
| 31 | \( 1 + 227552 T + p^{7} T^{2} \) |
| 37 | \( 1 + 160526 T + p^{7} T^{2} \) |
| 41 | \( 1 + 10842 T + p^{7} T^{2} \) |
| 43 | \( 1 + 630748 T + p^{7} T^{2} \) |
| 47 | \( 1 + 472656 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1494018 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2640660 T + p^{7} T^{2} \) |
| 61 | \( 1 - 827702 T + p^{7} T^{2} \) |
| 67 | \( 1 - 126004 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1414728 T + p^{7} T^{2} \) |
| 73 | \( 1 + 980282 T + p^{7} T^{2} \) |
| 79 | \( 1 + 3566800 T + p^{7} T^{2} \) |
| 83 | \( 1 + 5672892 T + p^{7} T^{2} \) |
| 89 | \( 1 - 11951190 T + p^{7} T^{2} \) |
| 97 | \( 1 + 8682146 T + p^{7} T^{2} \) |
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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
−9.746610323900973845929221721053, −9.214058044421978784169606051333, −7.83875535096920157374866104606, −6.80092325386742094856364294842, −5.77882889297338417001213472479, −5.21259867905917548975749469445, −3.43428918396002331413868064883, −2.97484543524493841640401803976, −1.62539192687212449331708170076, 0,
1.62539192687212449331708170076, 2.97484543524493841640401803976, 3.43428918396002331413868064883, 5.21259867905917548975749469445, 5.77882889297338417001213472479, 6.80092325386742094856364294842, 7.83875535096920157374866104606, 9.214058044421978784169606051333, 9.746610323900973845929221721053
