| L(s) = 1 | + 2-s + 4-s − 1.64·5-s + 2·7-s + 8-s − 1.64·10-s + 1.64·11-s + 5.29·13-s + 2·14-s + 16-s + 3.29·17-s + 0.354·19-s − 1.64·20-s + 1.64·22-s − 23-s − 2.29·25-s + 5.29·26-s + 2·28-s − 9.29·29-s − 1.29·31-s + 32-s + 3.29·34-s − 3.29·35-s + 6.93·37-s + 0.354·38-s − 1.64·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.736·5-s + 0.755·7-s + 0.353·8-s − 0.520·10-s + 0.496·11-s + 1.46·13-s + 0.534·14-s + 0.250·16-s + 0.798·17-s + 0.0812·19-s − 0.368·20-s + 0.350·22-s − 0.208·23-s − 0.458·25-s + 1.03·26-s + 0.377·28-s − 1.72·29-s − 0.231·31-s + 0.176·32-s + 0.564·34-s − 0.556·35-s + 1.14·37-s + 0.0574·38-s − 0.260·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.115921399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115921399\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 - 3.29T + 17T^{2} \) |
| 19 | \( 1 - 0.354T + 19T^{2} \) |
| 29 | \( 1 + 9.29T + 29T^{2} \) |
| 31 | \( 1 + 1.29T + 31T^{2} \) |
| 37 | \( 1 - 6.93T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 0.354T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.354T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 7.29T + 73T^{2} \) |
| 79 | \( 1 - 8.58T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 1.29T + 97T^{2} \) |
| 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
−11.43855571395339160704233000076, −10.68848734132717030147684526857, −9.373540314571360554561536372032, −8.213151869880932335026902257164, −7.58760820887045455896764147784, −6.34392027162519977867855247693, −5.39964865193001709794215659931, −4.15891181000549064408900710699, −3.43996159070286107653714567357, −1.57239268482570641614203851635,
1.57239268482570641614203851635, 3.43996159070286107653714567357, 4.15891181000549064408900710699, 5.39964865193001709794215659931, 6.34392027162519977867855247693, 7.58760820887045455896764147784, 8.213151869880932335026902257164, 9.373540314571360554561536372032, 10.68848734132717030147684526857, 11.43855571395339160704233000076
