| L(s) = 1 | + 5·5-s + 4·7-s − 48·11-s + 2·13-s + 114·17-s − 140·19-s + 72·23-s + 25·25-s − 210·29-s − 272·31-s + 20·35-s − 334·37-s + 198·41-s + 268·43-s + 216·47-s − 327·49-s + 78·53-s − 240·55-s + 240·59-s + 302·61-s + 10·65-s − 596·67-s − 768·71-s − 478·73-s − 192·77-s + 640·79-s − 348·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.215·7-s − 1.31·11-s + 0.0426·13-s + 1.62·17-s − 1.69·19-s + 0.652·23-s + 1/5·25-s − 1.34·29-s − 1.57·31-s + 0.0965·35-s − 1.48·37-s + 0.754·41-s + 0.950·43-s + 0.670·47-s − 0.953·49-s + 0.202·53-s − 0.588·55-s + 0.529·59-s + 0.633·61-s + 0.0190·65-s − 1.08·67-s − 1.28·71-s − 0.766·73-s − 0.284·77-s + 0.911·79-s − 0.460·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 140 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 210 T + p^{3} T^{2} \) |
| 31 | \( 1 + 272 T + p^{3} T^{2} \) |
| 37 | \( 1 + 334 T + p^{3} T^{2} \) |
| 41 | \( 1 - 198 T + p^{3} T^{2} \) |
| 43 | \( 1 - 268 T + p^{3} T^{2} \) |
| 47 | \( 1 - 216 T + p^{3} T^{2} \) |
| 53 | \( 1 - 78 T + p^{3} T^{2} \) |
| 59 | \( 1 - 240 T + p^{3} T^{2} \) |
| 61 | \( 1 - 302 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 + 478 T + p^{3} T^{2} \) |
| 79 | \( 1 - 640 T + p^{3} T^{2} \) |
| 83 | \( 1 + 348 T + p^{3} T^{2} \) |
| 89 | \( 1 + 210 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1534 T + p^{3} T^{2} \) |
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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
−9.619491182103650285765017499863, −8.730129701111845440422300778622, −7.82304087992908285675435985494, −7.07599125485653931229090053696, −5.74377227630537620006175915378, −5.29229770631922928062390085809, −3.96918257796561588452124102039, −2.76261014299176375902574960208, −1.64629280193516114942653260711, 0,
1.64629280193516114942653260711, 2.76261014299176375902574960208, 3.96918257796561588452124102039, 5.29229770631922928062390085809, 5.74377227630537620006175915378, 7.07599125485653931229090053696, 7.82304087992908285675435985494, 8.730129701111845440422300778622, 9.619491182103650285765017499863
