| L(s) = 1 | + 1.60·2-s − 1.72·3-s − 5.41·4-s − 2.78·6-s + 11.3·7-s − 21.5·8-s − 24.0·9-s − 40.6·11-s + 9.35·12-s + 12.3·13-s + 18.2·14-s + 8.55·16-s − 59.6·17-s − 38.6·18-s + 26.4·19-s − 19.5·21-s − 65.4·22-s − 107.·23-s + 37.2·24-s + 19.9·26-s + 88.1·27-s − 61.2·28-s + 173.·29-s − 332.·31-s + 186.·32-s + 70.2·33-s − 96.0·34-s + ⋯ |
| L(s) = 1 | + 0.568·2-s − 0.332·3-s − 0.676·4-s − 0.189·6-s + 0.611·7-s − 0.953·8-s − 0.889·9-s − 1.11·11-s + 0.224·12-s + 0.264·13-s + 0.347·14-s + 0.133·16-s − 0.851·17-s − 0.506·18-s + 0.318·19-s − 0.203·21-s − 0.633·22-s − 0.970·23-s + 0.317·24-s + 0.150·26-s + 0.628·27-s − 0.413·28-s + 1.11·29-s − 1.92·31-s + 1.02·32-s + 0.370·33-s − 0.484·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.077468228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.077468228\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + 47T \) |
| good | 2 | \( 1 - 1.60T + 8T^{2} \) |
| 3 | \( 1 + 1.72T + 27T^{2} \) |
| 7 | \( 1 - 11.3T + 343T^{2} \) |
| 11 | \( 1 + 40.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 332.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 172.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 178.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 63.2T + 7.95e4T^{2} \) |
| 53 | \( 1 - 402.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 305.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 86.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 681.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 726.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 79.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 279.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 556.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 342.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.63e3T + 9.12e5T^{2} \) |
| 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
−9.336855761062767642160437157799, −8.444212644849988570216417880677, −8.008367927951789945534216600230, −6.70202954299820221412985015474, −5.67016518152890794734071159370, −5.21967868114184782331678362622, −4.34691846346306906734917614577, −3.29455529101673026242604248372, −2.20621294529197593402827895188, −0.47964979178292421347621110341,
0.47964979178292421347621110341, 2.20621294529197593402827895188, 3.29455529101673026242604248372, 4.34691846346306906734917614577, 5.21967868114184782331678362622, 5.67016518152890794734071159370, 6.70202954299820221412985015474, 8.008367927951789945534216600230, 8.444212644849988570216417880677, 9.336855761062767642160437157799
