| L(s) = 1 | + 3-s − 4·5-s + 7-s + 9-s + 2·13-s − 4·15-s + 4·17-s + 3·19-s + 21-s + 2·23-s + 11·25-s + 27-s − 6·29-s − 5·31-s − 4·35-s − 3·37-s + 2·39-s − 2·41-s − 12·43-s − 4·45-s + 2·47-s − 6·49-s + 4·51-s − 6·53-s + 3·57-s + 10·59-s − 3·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s − 1.03·15-s + 0.970·17-s + 0.688·19-s + 0.218·21-s + 0.417·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 0.898·31-s − 0.676·35-s − 0.493·37-s + 0.320·39-s − 0.312·41-s − 1.82·43-s − 0.596·45-s + 0.291·47-s − 6/7·49-s + 0.560·51-s − 0.824·53-s + 0.397·57-s + 1.30·59-s − 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| 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
−15.77930612745963, −15.06288108150978, −14.78059733655565, −14.34147374556662, −13.56129532045419, −13.00284586321334, −12.46129908428407, −11.83740892751688, −11.43822345803531, −11.00782977923755, −10.28130323256447, −9.642962456400062, −8.848568533653025, −8.498850112299592, −7.855537261796946, −7.478391253782696, −7.037141893803127, −6.161221722292559, −5.180113178573374, −4.820802556250963, −3.833742227844273, −3.530527078677311, −3.059538511897290, −1.848986282585197, −1.059060739786115, 0,
1.059060739786115, 1.848986282585197, 3.059538511897290, 3.530527078677311, 3.833742227844273, 4.820802556250963, 5.180113178573374, 6.161221722292559, 7.037141893803127, 7.478391253782696, 7.855537261796946, 8.498850112299592, 8.848568533653025, 9.642962456400062, 10.28130323256447, 11.00782977923755, 11.43822345803531, 11.83740892751688, 12.46129908428407, 13.00284586321334, 13.56129532045419, 14.34147374556662, 14.78059733655565, 15.06288108150978, 15.77930612745963
