| L(s) = 1 | + 4.97·2-s + 9·3-s − 7.25·4-s + 44.7·6-s − 155.·7-s − 195.·8-s + 81·9-s + 121·11-s − 65.3·12-s − 1.17e3·13-s − 773.·14-s − 738.·16-s − 898.·17-s + 402.·18-s − 2.40e3·19-s − 1.39e3·21-s + 601.·22-s + 4.12e3·23-s − 1.75e3·24-s − 5.83e3·26-s + 729·27-s + 1.12e3·28-s + 75.4·29-s + 3.59e3·31-s + 2.57e3·32-s + 1.08e3·33-s − 4.46e3·34-s + ⋯ |
| L(s) = 1 | + 0.879·2-s + 0.577·3-s − 0.226·4-s + 0.507·6-s − 1.19·7-s − 1.07·8-s + 0.333·9-s + 0.301·11-s − 0.130·12-s − 1.92·13-s − 1.05·14-s − 0.721·16-s − 0.753·17-s + 0.293·18-s − 1.53·19-s − 0.692·21-s + 0.265·22-s + 1.62·23-s − 0.622·24-s − 1.69·26-s + 0.192·27-s + 0.271·28-s + 0.0166·29-s + 0.671·31-s + 0.444·32-s + 0.174·33-s − 0.662·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.877993157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.877993157\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 4.97T + 32T^{2} \) |
| 7 | \( 1 + 155.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 898.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.40e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 75.4T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.60e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.50e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.69e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.28e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 882.T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.87e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.34e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.09e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.312509619583299463189625670932, −8.953198032556497870953718119082, −7.64201127529653511233986221743, −6.70886090020869486944842262804, −6.00695904971009927797587180754, −4.67347074448276696818344583382, −4.24550407262949379977019551098, −2.93933573675537499323301673106, −2.49633227327153501593411835129, −0.49791808227505764499553871835,
0.49791808227505764499553871835, 2.49633227327153501593411835129, 2.93933573675537499323301673106, 4.24550407262949379977019551098, 4.67347074448276696818344583382, 6.00695904971009927797587180754, 6.70886090020869486944842262804, 7.64201127529653511233986221743, 8.953198032556497870953718119082, 9.312509619583299463189625670932
