| L(s) = 1 | − 2.56·3-s − 0.561·5-s + 3.56·9-s − 11-s + 2·13-s + 1.43·15-s + 7.12·17-s + 1.12·19-s + 7.68·23-s − 4.68·25-s − 1.43·27-s + 7.12·29-s + 5.43·31-s + 2.56·33-s − 5.68·37-s − 5.12·39-s − 8.24·41-s + 1.12·43-s − 2.00·45-s + 4·47-s − 7·49-s − 18.2·51-s + 8.24·53-s + 0.561·55-s − 2.87·57-s + 0.315·59-s + 9.36·61-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.251·5-s + 1.18·9-s − 0.301·11-s + 0.554·13-s + 0.371·15-s + 1.72·17-s + 0.257·19-s + 1.60·23-s − 0.936·25-s − 0.276·27-s + 1.32·29-s + 0.976·31-s + 0.445·33-s − 0.934·37-s − 0.820·39-s − 1.28·41-s + 0.171·43-s − 0.298·45-s + 0.583·47-s − 49-s − 2.55·51-s + 1.13·53-s + 0.0757·55-s − 0.381·57-s + 0.0410·59-s + 1.19·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8062143406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8062143406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 7.68T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 + 5.68T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 - 0.315T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 0.561T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
| 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
−11.71637930315705832383908896038, −10.55797848962274206480390322282, −10.06589824870708370937970806103, −8.639658734738239079781631052209, −7.52032751894912668633406699865, −6.51405201310665444503786570704, −5.57052142861751043992859836001, −4.80128707043152207413273055438, −3.30198380786633657028182233539, −1.00310502704205625408320475655,
1.00310502704205625408320475655, 3.30198380786633657028182233539, 4.80128707043152207413273055438, 5.57052142861751043992859836001, 6.51405201310665444503786570704, 7.52032751894912668633406699865, 8.639658734738239079781631052209, 10.06589824870708370937970806103, 10.55797848962274206480390322282, 11.71637930315705832383908896038
