| L(s) = 1 | − 4.75·2-s + 14.5·4-s + 7·7-s − 31.3·8-s − 7.31·11-s − 4.15·13-s − 33.2·14-s + 32.2·16-s − 53.5·17-s + 88.9·19-s + 34.7·22-s − 156.·23-s + 19.7·26-s + 102.·28-s − 42.2·29-s − 14.0·31-s + 97.4·32-s + 254.·34-s − 293.·37-s − 422.·38-s + 127.·41-s − 210.·43-s − 106.·44-s + 745.·46-s + 468.·47-s + 49·49-s − 60.6·52-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.82·4-s + 0.377·7-s − 1.38·8-s − 0.200·11-s − 0.0886·13-s − 0.635·14-s + 0.504·16-s − 0.763·17-s + 1.07·19-s + 0.337·22-s − 1.42·23-s + 0.148·26-s + 0.689·28-s − 0.270·29-s − 0.0812·31-s + 0.538·32-s + 1.28·34-s − 1.30·37-s − 1.80·38-s + 0.484·41-s − 0.745·43-s − 0.365·44-s + 2.38·46-s + 1.45·47-s + 0.142·49-s − 0.161·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7008306216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7008306216\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 4.75T + 8T^{2} \) |
| 11 | \( 1 + 7.31T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.15T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 156.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 42.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 14.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 127.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 210.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 468.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 115.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 314.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 768.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 717.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 737.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 477.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 279.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 776.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 29.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 231.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.970890811805694032396491088091, −8.413137318267117570905602137846, −7.61840104015083212704372299382, −7.06169990047794102396881656285, −6.08585292040275716448233716152, −5.06857525044980487288968828340, −3.83117405511955459300967708044, −2.48448647528250359621338834028, −1.67090541981498614793239259294, −0.52495703997550388500132063913,
0.52495703997550388500132063913, 1.67090541981498614793239259294, 2.48448647528250359621338834028, 3.83117405511955459300967708044, 5.06857525044980487288968828340, 6.08585292040275716448233716152, 7.06169990047794102396881656285, 7.61840104015083212704372299382, 8.413137318267117570905602137846, 8.970890811805694032396491088091
