| L(s) = 1 | + 0.517·2-s + 3-s − 1.73·4-s + 0.517·6-s − 1.93·8-s + 9-s + 3.46·11-s − 1.73·12-s − 4·13-s + 2.46·16-s − 4·17-s + 0.517·18-s + 5.27·19-s + 1.79·22-s − 3.48·23-s − 1.93·24-s − 2.07·26-s + 27-s − 4.92·29-s − 7.34·31-s + 5.13·32-s + 3.46·33-s − 2.07·34-s − 1.73·36-s − 10.5·37-s + 2.73·38-s − 4·39-s + ⋯ |
| L(s) = 1 | + 0.366·2-s + 0.577·3-s − 0.866·4-s + 0.211·6-s − 0.683·8-s + 0.333·9-s + 1.04·11-s − 0.500·12-s − 1.10·13-s + 0.616·16-s − 0.970·17-s + 0.122·18-s + 1.21·19-s + 0.382·22-s − 0.726·23-s − 0.394·24-s − 0.406·26-s + 0.192·27-s − 0.915·29-s − 1.31·31-s + 0.908·32-s + 0.603·33-s − 0.355·34-s − 0.288·36-s − 1.73·37-s + 0.443·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 0.517T + 2T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 + 0.757T + 59T^{2} \) |
| 61 | \( 1 - 0.656T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 6.39T + 71T^{2} \) |
| 73 | \( 1 - 2.92T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 18.9T + 97T^{2} \) |
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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
−8.288217180976412771977923695029, −7.38208564740232154935178596572, −6.82582154203810797856940145097, −5.69058849786341496225052905651, −5.09882552007965562050790716550, −4.12206433283912081163722844797, −3.70098669859013069107593474517, −2.65039203042174885673535125354, −1.54366958207981903076733913703, 0,
1.54366958207981903076733913703, 2.65039203042174885673535125354, 3.70098669859013069107593474517, 4.12206433283912081163722844797, 5.09882552007965562050790716550, 5.69058849786341496225052905651, 6.82582154203810797856940145097, 7.38208564740232154935178596572, 8.288217180976412771977923695029
