| L(s) = 1 | + 1.23·5-s + 2·7-s + 1.23·11-s + 4.47·13-s − 2.47·17-s + 3.23·19-s − 23-s − 3.47·25-s − 0.472·29-s + 10.4·31-s + 2.47·35-s − 5.70·37-s + 2·41-s + 0.763·43-s − 8.94·47-s − 3·49-s + 10.1·53-s + 1.52·55-s + 4·59-s + 1.70·61-s + 5.52·65-s − 5.70·67-s + 8.94·71-s − 4.47·73-s + 2.47·77-s + 14·79-s + 6.76·83-s + ⋯ |
| L(s) = 1 | + 0.552·5-s + 0.755·7-s + 0.372·11-s + 1.24·13-s − 0.599·17-s + 0.742·19-s − 0.208·23-s − 0.694·25-s − 0.0876·29-s + 1.88·31-s + 0.417·35-s − 0.938·37-s + 0.312·41-s + 0.116·43-s − 1.30·47-s − 0.428·49-s + 1.39·53-s + 0.206·55-s + 0.520·59-s + 0.218·61-s + 0.685·65-s − 0.697·67-s + 1.06·71-s − 0.523·73-s + 0.281·77-s + 1.57·79-s + 0.742·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.263410096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.263410096\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 6.76T + 83T^{2} \) |
| 89 | \( 1 + 0.944T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−9.366986654982897374702611334926, −8.506043931275713026481872721043, −7.982191341563105333220670617159, −6.84090187526359751429101500136, −6.15188299408633201902217535617, −5.31423566721584355027927066660, −4.39149665032998550353391823759, −3.43291891765510075536075244272, −2.14733361851274378675698918930, −1.15678218682883836901349598851,
1.15678218682883836901349598851, 2.14733361851274378675698918930, 3.43291891765510075536075244272, 4.39149665032998550353391823759, 5.31423566721584355027927066660, 6.15188299408633201902217535617, 6.84090187526359751429101500136, 7.982191341563105333220670617159, 8.506043931275713026481872721043, 9.366986654982897374702611334926
