| L(s) = 1 | − 0.399·2-s − 3-s − 1.84·4-s − 5-s + 0.399·6-s − 2.18·7-s + 1.53·8-s + 9-s + 0.399·10-s + 5.45·11-s + 1.84·12-s − 13-s + 0.870·14-s + 15-s + 3.06·16-s + 6.93·17-s − 0.399·18-s + 2.35·19-s + 1.84·20-s + 2.18·21-s − 2.17·22-s + 4.58·23-s − 1.53·24-s + 25-s + 0.399·26-s − 27-s + 4.01·28-s + ⋯ |
| L(s) = 1 | − 0.282·2-s − 0.577·3-s − 0.920·4-s − 0.447·5-s + 0.163·6-s − 0.824·7-s + 0.542·8-s + 0.333·9-s + 0.126·10-s + 1.64·11-s + 0.531·12-s − 0.277·13-s + 0.232·14-s + 0.258·15-s + 0.767·16-s + 1.68·17-s − 0.0941·18-s + 0.541·19-s + 0.411·20-s + 0.475·21-s − 0.464·22-s + 0.956·23-s − 0.313·24-s + 0.200·25-s + 0.0783·26-s − 0.192·27-s + 0.758·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.079917433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.079917433\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.399T + 2T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 - 5.45T + 11T^{2} \) |
| 17 | \( 1 - 6.93T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 - 4.58T + 23T^{2} \) |
| 29 | \( 1 + 0.133T + 29T^{2} \) |
| 37 | \( 1 - 8.94T + 37T^{2} \) |
| 41 | \( 1 + 0.596T + 41T^{2} \) |
| 43 | \( 1 + 7.42T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 + 1.10T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 + 7.44T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 1.69T + 73T^{2} \) |
| 79 | \( 1 + 0.227T + 79T^{2} \) |
| 83 | \( 1 + 3.26T + 83T^{2} \) |
| 89 | \( 1 - 10.8T + 89T^{2} \) |
| 97 | \( 1 - 6.37T + 97T^{2} \) |
| 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
−8.013236940608930026117167574896, −7.41273332966946371008716467907, −6.67301905562225788776145803562, −5.94337876269656683312224174015, −5.20189952614826823521591833330, −4.40950185214324922632708054872, −3.69395539126850332962216065802, −3.09207197970033750110732090567, −1.32558719887657900315138682056, −0.69194733333475171402580850668,
0.69194733333475171402580850668, 1.32558719887657900315138682056, 3.09207197970033750110732090567, 3.69395539126850332962216065802, 4.40950185214324922632708054872, 5.20189952614826823521591833330, 5.94337876269656683312224174015, 6.67301905562225788776145803562, 7.41273332966946371008716467907, 8.013236940608930026117167574896
