| L(s) = 1 | + 18.2·3-s + 25·5-s − 212.·7-s + 91.3·9-s − 704.·11-s + 169·13-s + 457.·15-s − 669.·17-s + 2.19e3·19-s − 3.88e3·21-s − 590.·23-s + 625·25-s − 2.77e3·27-s + 5.51e3·29-s + 4.29e3·31-s − 1.28e4·33-s − 5.30e3·35-s − 5.20e3·37-s + 3.09e3·39-s + 6.96e3·41-s − 831.·43-s + 2.28e3·45-s − 1.93e4·47-s + 2.82e4·49-s − 1.22e4·51-s − 3.16e4·53-s − 1.76e4·55-s + ⋯ |
| L(s) = 1 | + 1.17·3-s + 0.447·5-s − 1.63·7-s + 0.376·9-s − 1.75·11-s + 0.277·13-s + 0.524·15-s − 0.561·17-s + 1.39·19-s − 1.92·21-s − 0.232·23-s + 0.200·25-s − 0.731·27-s + 1.21·29-s + 0.802·31-s − 2.05·33-s − 0.732·35-s − 0.625·37-s + 0.325·39-s + 0.647·41-s − 0.0686·43-s + 0.168·45-s − 1.28·47-s + 1.68·49-s − 0.658·51-s − 1.54·53-s − 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.235216708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.235216708\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 13 | \( 1 - 169T \) |
| good | 3 | \( 1 - 18.2T + 243T^{2} \) |
| 7 | \( 1 + 212.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 704.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 669.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.19e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 590.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.20e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 831.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.93e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.71e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.86e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.41e5T + 8.58e9T^{2} \) |
| 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
−9.245002352551842378740124822212, −8.407275911460115145853186885371, −7.68524381682253999674926410883, −6.72294928287069959599229791945, −5.86820442915086425152236836921, −4.87631955589824635424818861004, −3.41529854011445098415398569955, −2.95675905563015880159725283566, −2.20313713927893227750615667049, −0.57264933634631720546309258187,
0.57264933634631720546309258187, 2.20313713927893227750615667049, 2.95675905563015880159725283566, 3.41529854011445098415398569955, 4.87631955589824635424818861004, 5.86820442915086425152236836921, 6.72294928287069959599229791945, 7.68524381682253999674926410883, 8.407275911460115145853186885371, 9.245002352551842378740124822212
