| L(s) = 1 | − 2-s + 4-s + 1.68·5-s − 3.55·7-s − 8-s − 1.68·10-s + 6.10·11-s + 1.47·13-s + 3.55·14-s + 16-s + 8.05·17-s − 4.84·19-s + 1.68·20-s − 6.10·22-s + 1.34·23-s − 2.16·25-s − 1.47·26-s − 3.55·28-s + 9.21·29-s + 10.1·31-s − 32-s − 8.05·34-s − 5.99·35-s + 6.09·37-s + 4.84·38-s − 1.68·40-s − 9.20·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.753·5-s − 1.34·7-s − 0.353·8-s − 0.532·10-s + 1.84·11-s + 0.408·13-s + 0.951·14-s + 0.250·16-s + 1.95·17-s − 1.11·19-s + 0.376·20-s − 1.30·22-s + 0.280·23-s − 0.432·25-s − 0.289·26-s − 0.672·28-s + 1.71·29-s + 1.82·31-s − 0.176·32-s − 1.38·34-s − 1.01·35-s + 1.00·37-s + 0.785·38-s − 0.266·40-s − 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.870472930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870472930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 - 1.68T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 11 | \( 1 - 6.10T + 11T^{2} \) |
| 13 | \( 1 - 1.47T + 13T^{2} \) |
| 17 | \( 1 - 8.05T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 - 1.34T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 + 9.20T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 + 8.87T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 + 3.25T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 - 2.77T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 8.87T + 83T^{2} \) |
| 89 | \( 1 - 1.15T + 89T^{2} \) |
| 97 | \( 1 + 8.68T + 97T^{2} \) |
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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
−8.107013216600489428533393808607, −6.84808076192935813265454136620, −6.37470794068726335339052821224, −6.27746087091484383426627001321, −5.21287241947822773305635186359, −4.07188655363814068207233124581, −3.37921218904252761964991862269, −2.66173015196483099036110423998, −1.50053338773089550767179249955, −0.824094314028197820386918929924,
0.824094314028197820386918929924, 1.50053338773089550767179249955, 2.66173015196483099036110423998, 3.37921218904252761964991862269, 4.07188655363814068207233124581, 5.21287241947822773305635186359, 6.27746087091484383426627001321, 6.37470794068726335339052821224, 6.84808076192935813265454136620, 8.107013216600489428533393808607
