| L(s) = 1 | − 3-s − 2·5-s + 9-s + 11-s + 13-s + 2·15-s + 2·17-s + 4·19-s − 25-s − 27-s − 2·29-s − 8·31-s − 33-s − 2·37-s − 39-s + 2·41-s − 4·43-s − 2·45-s + 8·47-s − 7·49-s − 2·51-s + 6·53-s − 2·55-s − 4·57-s − 4·59-s − 2·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.516·15-s + 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s − 49-s − 0.280·51-s + 0.824·53-s − 0.269·55-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.130409545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130409545\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−8.563632216758646801091997912902, −7.61591241186254996689820615805, −7.32140778602895878761348230626, −6.31386293576259429146441218657, −5.56951596418254229639769348565, −4.81923512911327994555871799591, −3.84339075470055414212716870701, −3.32831025650334139498794095432, −1.85760670391227356678888158189, −0.65482238861606688843692154746,
0.65482238861606688843692154746, 1.85760670391227356678888158189, 3.32831025650334139498794095432, 3.84339075470055414212716870701, 4.81923512911327994555871799591, 5.56951596418254229639769348565, 6.31386293576259429146441218657, 7.32140778602895878761348230626, 7.61591241186254996689820615805, 8.563632216758646801091997912902
