| L(s) = 1 | − 2·5-s − 5.12·11-s + 4.56·13-s + 3.12·17-s + 5.12·19-s + 23-s − 25-s + 0.561·29-s − 6.56·31-s − 8.24·37-s − 10.8·41-s − 8·43-s − 11.6·47-s − 7·49-s − 2·53-s + 10.2·55-s + 6.24·59-s + 12.2·61-s − 9.12·65-s − 5.12·67-s − 9.43·71-s − 2.31·73-s − 5.12·79-s + 2.24·83-s − 6.24·85-s + 13.3·89-s − 10.2·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.54·11-s + 1.26·13-s + 0.757·17-s + 1.17·19-s + 0.208·23-s − 0.200·25-s + 0.104·29-s − 1.17·31-s − 1.35·37-s − 1.68·41-s − 1.21·43-s − 1.70·47-s − 49-s − 0.274·53-s + 1.38·55-s + 0.813·59-s + 1.56·61-s − 1.13·65-s − 0.625·67-s − 1.12·71-s − 0.270·73-s − 0.576·79-s + 0.246·83-s − 0.677·85-s + 1.41·89-s − 1.05·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 - 3.12T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 29 | \( 1 - 0.561T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 6.24T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 + 9.43T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.746736385064706737840619030030, −8.124290424184138514192108824047, −7.55348928553054491057742265065, −6.67963513332516710599777904649, −5.48826647517209078756682284807, −5.00415181104012127363492075101, −3.59820751860751599702084420667, −3.20717455827987912561325873685, −1.59770949758107538041436817562, 0,
1.59770949758107538041436817562, 3.20717455827987912561325873685, 3.59820751860751599702084420667, 5.00415181104012127363492075101, 5.48826647517209078756682284807, 6.67963513332516710599777904649, 7.55348928553054491057742265065, 8.124290424184138514192108824047, 8.746736385064706737840619030030
