| L(s) = 1 | + 1.73·2-s + 0.999·4-s + 5-s + 2·7-s − 1.73·8-s + 1.73·10-s + 4.73·11-s + 13-s + 3.46·14-s − 5·16-s + 3.46·17-s − 6.19·19-s + 0.999·20-s + 8.19·22-s − 1.26·23-s + 25-s + 1.73·26-s + 1.99·28-s + 2.53·29-s + 10.1·31-s − 5.19·32-s + 5.99·34-s + 2·35-s − 4·37-s − 10.7·38-s − 1.73·40-s − 3.46·41-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 0.447·5-s + 0.755·7-s − 0.612·8-s + 0.547·10-s + 1.42·11-s + 0.277·13-s + 0.925·14-s − 1.25·16-s + 0.840·17-s − 1.42·19-s + 0.223·20-s + 1.74·22-s − 0.264·23-s + 0.200·25-s + 0.339·26-s + 0.377·28-s + 0.470·29-s + 1.83·31-s − 0.918·32-s + 1.02·34-s + 0.338·35-s − 0.657·37-s − 1.74·38-s − 0.273·40-s − 0.541·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.984453741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.984453741\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 6.19T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 0.196T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 9.12T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 - 6.39T + 67T^{2} \) |
| 71 | \( 1 + 4.73T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.94472824657291446880371885590, −9.877348407940449699969248205996, −8.904475350883832398689412944219, −8.113862234208344136020896150558, −6.56456973101159506347002258038, −6.15119478756246662288907107930, −4.92456319302782996777442433173, −4.25102037691181755072506709673, −3.13978927529927978171053329719, −1.63717590887318719464495311003,
1.63717590887318719464495311003, 3.13978927529927978171053329719, 4.25102037691181755072506709673, 4.92456319302782996777442433173, 6.15119478756246662288907107930, 6.56456973101159506347002258038, 8.113862234208344136020896150558, 8.904475350883832398689412944219, 9.877348407940449699969248205996, 10.94472824657291446880371885590
