| L(s) = 1 | − 2-s + 4-s + 2.73·5-s − 2·7-s − 8-s − 2.73·10-s − 4.46·11-s − 13-s + 2·14-s + 16-s − 0.464·17-s + 0.464·19-s + 2.73·20-s + 4.46·22-s + 6.19·23-s + 2.46·25-s + 26-s − 2·28-s + 5.26·29-s − 10.1·31-s − 32-s + 0.464·34-s − 5.46·35-s − 6.92·37-s − 0.464·38-s − 2.73·40-s − 7.73·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.22·5-s − 0.755·7-s − 0.353·8-s − 0.863·10-s − 1.34·11-s − 0.277·13-s + 0.534·14-s + 0.250·16-s − 0.112·17-s + 0.106·19-s + 0.610·20-s + 0.951·22-s + 1.29·23-s + 0.492·25-s + 0.196·26-s − 0.377·28-s + 0.978·29-s − 1.83·31-s − 0.176·32-s + 0.0795·34-s − 0.923·35-s − 1.13·37-s − 0.0752·38-s − 0.431·40-s − 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4.46T + 11T^{2} \) |
| 17 | \( 1 + 0.464T + 17T^{2} \) |
| 19 | \( 1 - 0.464T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 5.26T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 + 0.267T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 + 0.732T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 7.66T + 71T^{2} \) |
| 73 | \( 1 - 4.66T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 6.92T + 89T^{2} \) |
| 97 | \( 1 + 9.19T + 97T^{2} \) |
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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
−8.888399566827682293847111536135, −8.046496172914796232197311663675, −7.05977301449307605435886629225, −6.55186786807745228411426794403, −5.50734013852034458376185153821, −5.05555206481506980585097761391, −3.35831926832297222844458032021, −2.59127458746018617076457123581, −1.62573860093534427498483262336, 0,
1.62573860093534427498483262336, 2.59127458746018617076457123581, 3.35831926832297222844458032021, 5.05555206481506980585097761391, 5.50734013852034458376185153821, 6.55186786807745228411426794403, 7.05977301449307605435886629225, 8.046496172914796232197311663675, 8.888399566827682293847111536135
