| L(s) = 1 | + 3.36·3-s + 5-s + 1.19·7-s + 8.29·9-s − 2.16·11-s + 13-s + 3.36·15-s + 6·17-s − 2.16·19-s + 4·21-s − 7.70·23-s + 25-s + 17.7·27-s + 5.29·29-s − 4.55·31-s − 7.29·33-s + 1.19·35-s − 9.29·37-s + 3.36·39-s + 12.5·41-s + 10.0·43-s + 8.29·45-s + 1.19·47-s − 5.58·49-s + 20.1·51-s + 8.58·53-s − 2.16·55-s + ⋯ |
| L(s) = 1 | + 1.94·3-s + 0.447·5-s + 0.449·7-s + 2.76·9-s − 0.654·11-s + 0.277·13-s + 0.867·15-s + 1.45·17-s − 0.497·19-s + 0.872·21-s − 1.60·23-s + 0.200·25-s + 3.42·27-s + 0.982·29-s − 0.817·31-s − 1.26·33-s + 0.201·35-s − 1.52·37-s + 0.538·39-s + 1.96·41-s + 1.53·43-s + 1.23·45-s + 0.173·47-s − 0.797·49-s + 2.82·51-s + 1.17·53-s − 0.292·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.870616779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.870616779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 3.36T + 3T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 4.55T + 31T^{2} \) |
| 37 | \( 1 + 9.29T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 + 2.70T + 61T^{2} \) |
| 67 | \( 1 + 5.53T + 67T^{2} \) |
| 71 | \( 1 - 6.50T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 - 4.33T + 79T^{2} \) |
| 83 | \( 1 - 1.19T + 83T^{2} \) |
| 89 | \( 1 + 4.58T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−8.346840667044311195966115618374, −7.79847879090087303237302001555, −7.39611728991401903493319150186, −6.28253476795186545645862975854, −5.39158544571761660569391187527, −4.36294999114558105211173905018, −3.70531691285854824896158486938, −2.82809358308422103685722007854, −2.14693831598345330449590498533, −1.29601947823929830137500393836,
1.29601947823929830137500393836, 2.14693831598345330449590498533, 2.82809358308422103685722007854, 3.70531691285854824896158486938, 4.36294999114558105211173905018, 5.39158544571761660569391187527, 6.28253476795186545645862975854, 7.39611728991401903493319150186, 7.79847879090087303237302001555, 8.346840667044311195966115618374
