| L(s) = 1 | − 12·3-s − 210·5-s − 1.01e3·7-s − 2.04e3·9-s − 1.09e3·11-s + 1.38e3·13-s + 2.52e3·15-s + 1.47e4·17-s + 3.99e4·19-s + 1.21e4·21-s − 6.87e4·23-s − 3.40e4·25-s + 5.07e4·27-s − 1.02e5·29-s − 2.27e5·31-s + 1.31e4·33-s + 2.13e5·35-s + 1.60e5·37-s − 1.65e4·39-s + 1.08e4·41-s + 6.30e5·43-s + 4.29e5·45-s − 4.72e5·47-s + 2.08e5·49-s − 1.76e5·51-s − 1.49e6·53-s + 2.29e5·55-s + ⋯ |
| L(s) = 1 | − 0.256·3-s − 0.751·5-s − 1.11·7-s − 0.934·9-s − 0.247·11-s + 0.174·13-s + 0.192·15-s + 0.725·17-s + 1.33·19-s + 0.287·21-s − 1.17·23-s − 0.435·25-s + 0.496·27-s − 0.780·29-s − 1.37·31-s + 0.0634·33-s + 0.841·35-s + 0.521·37-s − 0.0447·39-s + 0.0245·41-s + 1.20·43-s + 0.701·45-s − 0.664·47-s + 0.253·49-s − 0.186·51-s − 1.37·53-s + 0.185·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 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 | \( 1 + 1092 T + p^{7} T^{2} \) |
| 13 | \( 1 - 1382 T + p^{7} T^{2} \) |
| 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
−16.61230088559662560403243379778, −15.75840624968036963043557726119, −14.12355208363710712333207539217, −12.52030974291653514059742068362, −11.30617514104106319868470141387, −9.565435243058661987926040580078, −7.72947728406922360183810037330, −5.82525239813900472703591586342, −3.39930778928952202693084202511, 0,
3.39930778928952202693084202511, 5.82525239813900472703591586342, 7.72947728406922360183810037330, 9.565435243058661987926040580078, 11.30617514104106319868470141387, 12.52030974291653514059742068362, 14.12355208363710712333207539217, 15.75840624968036963043557726119, 16.61230088559662560403243379778
