| L(s) = 1 | − 5-s − 1.87·7-s + 3.38·11-s + 0.553·13-s + 2.90·17-s + 1.62·19-s + 0.389·23-s + 25-s − 8.75·29-s + 4.37·31-s + 1.87·35-s − 37-s + 5.69·41-s + 5.18·43-s − 6.57·47-s − 3.47·49-s + 2.04·53-s − 3.38·55-s − 1.54·59-s − 6.85·61-s − 0.553·65-s + 13.3·67-s − 0.493·71-s − 10.2·73-s − 6.35·77-s − 11.5·79-s + 10.8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.709·7-s + 1.02·11-s + 0.153·13-s + 0.705·17-s + 0.372·19-s + 0.0811·23-s + 0.200·25-s − 1.62·29-s + 0.786·31-s + 0.317·35-s − 0.164·37-s + 0.889·41-s + 0.790·43-s − 0.959·47-s − 0.496·49-s + 0.281·53-s − 0.456·55-s − 0.201·59-s − 0.878·61-s − 0.0685·65-s + 1.62·67-s − 0.0586·71-s − 1.19·73-s − 0.724·77-s − 1.29·79-s + 1.18·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684050202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684050202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 - 3.38T + 11T^{2} \) |
| 13 | \( 1 - 0.553T + 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 - 0.389T + 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 - 4.37T + 31T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 - 5.18T + 43T^{2} \) |
| 47 | \( 1 + 6.57T + 47T^{2} \) |
| 53 | \( 1 - 2.04T + 53T^{2} \) |
| 59 | \( 1 + 1.54T + 59T^{2} \) |
| 61 | \( 1 + 6.85T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 0.493T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.84186485417084402681637377985, −7.36970125728789793250201158634, −6.52995095931044306094465851873, −6.01430070884182393868890576156, −5.17027363211347575685519167232, −4.22228643219623533726363062038, −3.59657998065538710802836138738, −2.95116477985669005253790115023, −1.72947392771404935530097599635, −0.68186415576831683828927508557,
0.68186415576831683828927508557, 1.72947392771404935530097599635, 2.95116477985669005253790115023, 3.59657998065538710802836138738, 4.22228643219623533726363062038, 5.17027363211347575685519167232, 6.01430070884182393868890576156, 6.52995095931044306094465851873, 7.36970125728789793250201158634, 7.84186485417084402681637377985
