| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s + 2·11-s + 4·13-s − 15-s − 6·17-s − 19-s − 21-s − 6·23-s − 4·25-s + 5·27-s − 7·29-s − 2·31-s − 2·33-s + 35-s + 2·37-s − 4·39-s + 6·41-s + 6·43-s − 2·45-s + 47-s + 49-s + 6·51-s + 2·55-s + 57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s − 0.218·21-s − 1.25·23-s − 4/5·25-s + 0.962·27-s − 1.29·29-s − 0.359·31-s − 0.348·33-s + 0.169·35-s + 0.328·37-s − 0.640·39-s + 0.937·41-s + 0.914·43-s − 0.298·45-s + 0.145·47-s + 1/7·49-s + 0.840·51-s + 0.269·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−13.99369218027643, −13.55059365284728, −13.04868776834304, −12.55353392037402, −11.89490194764440, −11.41705770707108, −11.19607956020717, −10.68082236318332, −10.21128212311022, −9.406925975791239, −8.987575565428963, −8.749619259108616, −7.914533708089427, −7.597924183309416, −6.736139074976686, −6.238126809926264, −5.875087509843079, −5.591135294425248, −4.655038252396157, −4.213528098367691, −3.716994095197134, −2.916062397852739, −2.031604518865628, −1.805247746884937, −0.7940724262743462, 0,
0.7940724262743462, 1.805247746884937, 2.031604518865628, 2.916062397852739, 3.716994095197134, 4.213528098367691, 4.655038252396157, 5.591135294425248, 5.875087509843079, 6.238126809926264, 6.736139074976686, 7.597924183309416, 7.914533708089427, 8.749619259108616, 8.987575565428963, 9.406925975791239, 10.21128212311022, 10.68082236318332, 11.19607956020717, 11.41705770707108, 11.89490194764440, 12.55353392037402, 13.04868776834304, 13.55059365284728, 13.99369218027643
