| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 11-s + 13-s − 14-s − 16-s − 2·17-s − 4·19-s + 22-s − 26-s − 28-s − 6·29-s − 4·31-s − 5·32-s + 2·34-s − 6·37-s + 4·38-s − 6·41-s − 12·43-s + 44-s + 4·47-s + 49-s − 52-s − 10·53-s + 3·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 0.301·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.213·22-s − 0.196·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s − 0.986·37-s + 0.648·38-s − 0.937·41-s − 1.82·43-s + 0.150·44-s + 0.583·47-s + 1/7·49-s − 0.138·52-s − 1.37·53-s + 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.37376660322465, −13.12112394505443, −12.49125700426358, −12.03650593311393, −11.27266650838385, −11.13905264959494, −10.38699560160874, −10.29468512035910, −9.623806177376759, −9.125110213921768, −8.588014027586035, −8.499777091400054, −7.831967354586216, −7.352624672614723, −6.912762593949651, −6.310054614883206, −5.651640427767284, −5.179672805813744, −4.669244129942914, −4.214173953354565, −3.574241752310797, −3.118590661345897, −1.980103258111357, −1.890109254923762, −1.121792305950790, 0, 0,
1.121792305950790, 1.890109254923762, 1.980103258111357, 3.118590661345897, 3.574241752310797, 4.214173953354565, 4.669244129942914, 5.179672805813744, 5.651640427767284, 6.310054614883206, 6.912762593949651, 7.352624672614723, 7.831967354586216, 8.499777091400054, 8.588014027586035, 9.125110213921768, 9.623806177376759, 10.29468512035910, 10.38699560160874, 11.13905264959494, 11.27266650838385, 12.03650593311393, 12.49125700426358, 13.12112394505443, 13.37376660322465
