| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 4·7-s + 8-s + 9-s − 2·11-s + 12-s + 4·13-s + 4·14-s + 16-s + 2·17-s + 18-s − 7·19-s + 4·21-s − 2·22-s + 3·23-s + 24-s + 4·26-s + 27-s + 4·28-s − 8·31-s + 32-s − 2·33-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s + 1.10·13-s + 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.60·19-s + 0.872·21-s − 0.426·22-s + 0.625·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.755·28-s − 1.43·31-s + 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.051672660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.051672660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.995296574738187782154599674742, −7.61683793157609303950027771103, −6.72674888985966770528166868382, −5.77325655820792269986064550668, −5.28035230204123632159795211171, −4.25556228057089052710699668776, −3.97015755616311651646126320790, −2.76110446186324297039618712855, −2.05508181228896234272145846513, −1.14892703587670228042642546981,
1.14892703587670228042642546981, 2.05508181228896234272145846513, 2.76110446186324297039618712855, 3.97015755616311651646126320790, 4.25556228057089052710699668776, 5.28035230204123632159795211171, 5.77325655820792269986064550668, 6.72674888985966770528166868382, 7.61683793157609303950027771103, 7.995296574738187782154599674742
