| L(s) = 1 | − 1.29·2-s − 0.335·4-s + 5-s + 2.78·7-s + 3.01·8-s − 1.29·10-s + 11-s − 4.84·13-s − 3.58·14-s − 3.21·16-s + 7.95·17-s + 19-s − 0.335·20-s − 1.29·22-s + 5.02·23-s + 25-s + 6.25·26-s − 0.932·28-s + 1.85·29-s + 7.93·31-s − 1.87·32-s − 10.2·34-s + 2.78·35-s + 5.10·37-s − 1.29·38-s + 3.01·40-s + 2.69·41-s + ⋯ |
| L(s) = 1 | − 0.912·2-s − 0.167·4-s + 0.447·5-s + 1.05·7-s + 1.06·8-s − 0.408·10-s + 0.301·11-s − 1.34·13-s − 0.959·14-s − 0.804·16-s + 1.92·17-s + 0.229·19-s − 0.0749·20-s − 0.275·22-s + 1.04·23-s + 0.200·25-s + 1.22·26-s − 0.176·28-s + 0.344·29-s + 1.42·31-s − 0.331·32-s − 1.76·34-s + 0.470·35-s + 0.838·37-s − 0.209·38-s + 0.476·40-s + 0.421·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664730724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664730724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 13 | \( 1 + 4.84T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 23 | \( 1 - 5.02T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 - 7.93T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 - 2.69T + 41T^{2} \) |
| 43 | \( 1 + 7.01T + 43T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 + 0.915T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 5.03T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 8.17T + 83T^{2} \) |
| 89 | \( 1 + 7.60T + 89T^{2} \) |
| 97 | \( 1 - 2.16T + 97T^{2} \) |
| 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
−7.897186743749008196837237088062, −7.29846264383960748206082165653, −6.53376024563200856035630046301, −5.45143283616616268654473937025, −4.97709949650497742560909932698, −4.45556348716624405235988277269, −3.30687552872682053216923006083, −2.39383462465447327571752502344, −1.40534835004518902459365597833, −0.815591746329266581177900179484,
0.815591746329266581177900179484, 1.40534835004518902459365597833, 2.39383462465447327571752502344, 3.30687552872682053216923006083, 4.45556348716624405235988277269, 4.97709949650497742560909932698, 5.45143283616616268654473937025, 6.53376024563200856035630046301, 7.29846264383960748206082165653, 7.897186743749008196837237088062
