| L(s) = 1 | − 2-s + 4-s + 1.09·5-s − 1.25·7-s − 8-s − 1.09·10-s − 0.173·11-s + 4.22·13-s + 1.25·14-s + 16-s − 4.61·17-s + 3.05·19-s + 1.09·20-s + 0.173·22-s − 0.515·23-s − 3.80·25-s − 4.22·26-s − 1.25·28-s + 5.77·29-s + 0.0135·31-s − 32-s + 4.61·34-s − 1.36·35-s − 7.73·37-s − 3.05·38-s − 1.09·40-s − 7.34·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.488·5-s − 0.473·7-s − 0.353·8-s − 0.345·10-s − 0.0523·11-s + 1.17·13-s + 0.334·14-s + 0.250·16-s − 1.11·17-s + 0.700·19-s + 0.244·20-s + 0.0370·22-s − 0.107·23-s − 0.761·25-s − 0.829·26-s − 0.236·28-s + 1.07·29-s + 0.00243·31-s − 0.176·32-s + 0.791·34-s − 0.231·35-s − 1.27·37-s − 0.494·38-s − 0.172·40-s − 1.14·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.09T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 0.173T + 11T^{2} \) |
| 13 | \( 1 - 4.22T + 13T^{2} \) |
| 17 | \( 1 + 4.61T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 0.515T + 23T^{2} \) |
| 29 | \( 1 - 5.77T + 29T^{2} \) |
| 31 | \( 1 - 0.0135T + 31T^{2} \) |
| 37 | \( 1 + 7.73T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 1.74T + 53T^{2} \) |
| 59 | \( 1 + 8.36T + 59T^{2} \) |
| 61 | \( 1 - 6.66T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 5.70T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 1.27T + 79T^{2} \) |
| 83 | \( 1 - 4.98T + 83T^{2} \) |
| 89 | \( 1 + 3.60T + 89T^{2} \) |
| 97 | \( 1 + 7.13T + 97T^{2} \) |
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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
−7.52574807497405097638243768812, −6.69934794039000549622509656364, −6.31537206785508788998158036905, −5.59894805468081705249989817834, −4.71581966959371302879767328360, −3.71575938562277225007221687544, −3.02133343725273909712354167311, −2.05075823063855764280234900746, −1.25897381282847814241757664138, 0,
1.25897381282847814241757664138, 2.05075823063855764280234900746, 3.02133343725273909712354167311, 3.71575938562277225007221687544, 4.71581966959371302879767328360, 5.59894805468081705249989817834, 6.31537206785508788998158036905, 6.69934794039000549622509656364, 7.52574807497405097638243768812
