| L(s) = 1 | − 1.67·2-s + 0.799·4-s + 5-s − 2.20·7-s + 2.00·8-s − 1.67·10-s − 5.87·11-s + 3.68·14-s − 4.95·16-s − 2.98·17-s − 3.00·19-s + 0.799·20-s + 9.83·22-s + 3.33·23-s + 25-s − 1.75·28-s + 3.00·29-s + 7.61·31-s + 4.28·32-s + 4.99·34-s − 2.20·35-s − 10.5·37-s + 5.02·38-s + 2.00·40-s − 0.0937·41-s + 5.55·43-s − 4.69·44-s + ⋯ |
| L(s) = 1 | − 1.18·2-s + 0.399·4-s + 0.447·5-s − 0.831·7-s + 0.710·8-s − 0.529·10-s − 1.77·11-s + 0.984·14-s − 1.23·16-s − 0.724·17-s − 0.688·19-s + 0.178·20-s + 2.09·22-s + 0.694·23-s + 0.200·25-s − 0.332·28-s + 0.558·29-s + 1.36·31-s + 0.756·32-s + 0.856·34-s − 0.372·35-s − 1.73·37-s + 0.815·38-s + 0.317·40-s − 0.0146·41-s + 0.846·43-s − 0.708·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3707960048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3707960048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 + 5.87T + 11T^{2} \) |
| 17 | \( 1 + 2.98T + 17T^{2} \) |
| 19 | \( 1 + 3.00T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 - 7.61T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 0.0937T + 41T^{2} \) |
| 43 | \( 1 - 5.55T + 43T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 - 7.52T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 8.88T + 71T^{2} \) |
| 73 | \( 1 - 4.62T + 73T^{2} \) |
| 79 | \( 1 + 9.51T + 79T^{2} \) |
| 83 | \( 1 - 3.47T + 83T^{2} \) |
| 89 | \( 1 - 4.95T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.022133372399812282444819639305, −7.33493487654132367085310251892, −6.64467284153024669166807738050, −6.00922892574294876328083037686, −4.96187800890748211207180276496, −4.55913719295937000515139688858, −3.19517492167119181780778572350, −2.55150042350866683232229003906, −1.64050475786766580393042904390, −0.35805038150387670495079434173,
0.35805038150387670495079434173, 1.64050475786766580393042904390, 2.55150042350866683232229003906, 3.19517492167119181780778572350, 4.55913719295937000515139688858, 4.96187800890748211207180276496, 6.00922892574294876328083037686, 6.64467284153024669166807738050, 7.33493487654132367085310251892, 8.022133372399812282444819639305
