| L(s) = 1 | − 2.81·2-s + 5.93·4-s − 11.0·8-s − 2.72·11-s + 19.3·16-s + 7.67·22-s + 9.55·23-s − 5·25-s − 8.16·29-s − 32.4·32-s + 8.98·37-s − 13.1·43-s − 16.1·44-s − 26.9·46-s + 14.0·50-s + 5.19·53-s + 22.9·58-s + 52.6·64-s + 14.9·67-s + 11.7·71-s − 25.3·74-s − 14.0·79-s + 36.9·86-s + 30.2·88-s + 56.7·92-s − 29.6·100-s − 14.6·106-s + ⋯ |
| L(s) = 1 | − 1.99·2-s + 2.96·4-s − 3.92·8-s − 0.820·11-s + 4.84·16-s + 1.63·22-s + 1.99·23-s − 25-s − 1.51·29-s − 5.73·32-s + 1.47·37-s − 1.99·43-s − 2.43·44-s − 3.96·46-s + 1.99·50-s + 0.713·53-s + 3.01·58-s + 6.57·64-s + 1.82·67-s + 1.38·71-s − 2.94·74-s − 1.58·79-s + 3.98·86-s + 3.22·88-s + 5.91·92-s − 2.96·100-s − 1.42·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5758798910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5758798910\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 2.72T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 9.55T + 23T^{2} \) |
| 29 | \( 1 + 8.16T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 13.1T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.670997123163164472973451190893, −8.102971767777946389251159909174, −7.39892404614494501230063067944, −6.87378494843589526366146912377, −5.95224954186248681964499535855, −5.18496653344924388881137988957, −3.52458922955635854507724263958, −2.64996134390781274069124199661, −1.78212994970985780484705625593, −0.60827202986720964015324917781,
0.60827202986720964015324917781, 1.78212994970985780484705625593, 2.64996134390781274069124199661, 3.52458922955635854507724263958, 5.18496653344924388881137988957, 5.95224954186248681964499535855, 6.87378494843589526366146912377, 7.39892404614494501230063067944, 8.102971767777946389251159909174, 8.670997123163164472973451190893
