| L(s) = 1 | − 0.523·3-s − 5-s + 0.476·7-s − 2.72·9-s − 1.67·11-s + 2.67·13-s + 0.523·15-s + 1.67·17-s + 7.92·19-s − 0.249·21-s + 23-s + 25-s + 3·27-s − 2.20·29-s + 6.77·31-s + 0.878·33-s − 0.476·35-s + 4·37-s − 1.40·39-s + 5.97·41-s − 0.402·43-s + 2.72·45-s + 1.79·47-s − 6.77·49-s − 0.878·51-s − 10.8·53-s + 1.67·55-s + ⋯ |
| L(s) = 1 | − 0.302·3-s − 0.447·5-s + 0.179·7-s − 0.908·9-s − 0.505·11-s + 0.742·13-s + 0.135·15-s + 0.406·17-s + 1.81·19-s − 0.0544·21-s + 0.208·23-s + 0.200·25-s + 0.577·27-s − 0.408·29-s + 1.21·31-s + 0.153·33-s − 0.0804·35-s + 0.657·37-s − 0.224·39-s + 0.933·41-s − 0.0614·43-s + 0.406·45-s + 0.262·47-s − 0.967·49-s − 0.123·51-s − 1.48·53-s + 0.226·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.258313130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258313130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.523T + 3T^{2} \) |
| 7 | \( 1 - 0.476T + 7T^{2} \) |
| 11 | \( 1 + 1.67T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 29 | \( 1 + 2.20T + 29T^{2} \) |
| 31 | \( 1 - 6.77T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 + 0.402T + 43T^{2} \) |
| 47 | \( 1 - 1.79T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 9.45T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 - 9.97T + 71T^{2} \) |
| 73 | \( 1 + 6.10T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 + 6.08T + 97T^{2} \) |
| 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
−10.08984429395356211569474448110, −9.268086343846071693191280694678, −8.227255071204510559461661522701, −7.74931592723668786958483191954, −6.60294314336283483139283473760, −5.64340705669167632379914834182, −4.95667437990707476866512516339, −3.64365954942181291161219536478, −2.73648436186003824146470648247, −0.925421902208183046618648672404,
0.925421902208183046618648672404, 2.73648436186003824146470648247, 3.64365954942181291161219536478, 4.95667437990707476866512516339, 5.64340705669167632379914834182, 6.60294314336283483139283473760, 7.74931592723668786958483191954, 8.227255071204510559461661522701, 9.268086343846071693191280694678, 10.08984429395356211569474448110
