| L(s) = 1 | + 2.05·5-s + 2.46·7-s − 1.46·11-s − 3.64·13-s − 5.92·17-s − 19-s + 0.320·23-s − 0.780·25-s − 0.945·29-s + 7.19·31-s + 5.05·35-s + 2.59·37-s − 6.86·41-s − 5.78·43-s + 1.81·47-s − 0.945·49-s − 5.96·53-s − 3·55-s + 11.2·59-s + 2.83·61-s − 7.49·65-s − 6.62·67-s − 2.64·71-s + 3.81·73-s − 3.59·77-s − 7.24·79-s + 0.0789·83-s + ⋯ |
| L(s) = 1 | + 0.918·5-s + 0.929·7-s − 0.440·11-s − 1.01·13-s − 1.43·17-s − 0.229·19-s + 0.0667·23-s − 0.156·25-s − 0.175·29-s + 1.29·31-s + 0.854·35-s + 0.426·37-s − 1.07·41-s − 0.882·43-s + 0.264·47-s − 0.135·49-s − 0.819·53-s − 0.404·55-s + 1.46·59-s + 0.362·61-s − 0.929·65-s − 0.809·67-s − 0.314·71-s + 0.446·73-s − 0.409·77-s − 0.815·79-s + 0.00867·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.05T + 5T^{2} \) |
| 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 3.64T + 13T^{2} \) |
| 17 | \( 1 + 5.92T + 17T^{2} \) |
| 23 | \( 1 - 0.320T + 23T^{2} \) |
| 29 | \( 1 + 0.945T + 29T^{2} \) |
| 31 | \( 1 - 7.19T + 31T^{2} \) |
| 37 | \( 1 - 2.59T + 37T^{2} \) |
| 41 | \( 1 + 6.86T + 41T^{2} \) |
| 43 | \( 1 + 5.78T + 43T^{2} \) |
| 47 | \( 1 - 1.81T + 47T^{2} \) |
| 53 | \( 1 + 5.96T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 2.83T + 61T^{2} \) |
| 67 | \( 1 + 6.62T + 67T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 - 3.81T + 73T^{2} \) |
| 79 | \( 1 + 7.24T + 79T^{2} \) |
| 83 | \( 1 - 0.0789T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.44154133495018095218604097166, −6.75074212694282657464961369247, −6.11660233270263802109012614044, −5.22450398333177698952108081875, −4.81965291957982615350373218591, −4.08672003458890502993284054091, −2.78869595509198631770418664205, −2.21836995665255540429472968939, −1.47763921002209922285288479761, 0,
1.47763921002209922285288479761, 2.21836995665255540429472968939, 2.78869595509198631770418664205, 4.08672003458890502993284054091, 4.81965291957982615350373218591, 5.22450398333177698952108081875, 6.11660233270263802109012614044, 6.75074212694282657464961369247, 7.44154133495018095218604097166
