| L(s) = 1 | + (0.873 − 1.49i)3-s + (−0.268 − 0.738i)5-s + (−3.49 + 0.616i)7-s + (−1.47 − 2.61i)9-s + (−4.77 − 1.73i)11-s + (−4.34 − 3.64i)13-s + (−1.33 − 0.243i)15-s + (6.45 + 3.72i)17-s + (2.31 − 1.33i)19-s + (−2.13 + 5.77i)21-s + (0.749 − 4.25i)23-s + (3.35 − 2.81i)25-s + (−5.19 − 0.0767i)27-s + (3.09 + 3.68i)29-s + (−2.76 − 0.487i)31-s + ⋯ |
| L(s) = 1 | + (0.504 − 0.863i)3-s + (−0.120 − 0.330i)5-s + (−1.32 + 0.233i)7-s + (−0.491 − 0.870i)9-s + (−1.43 − 0.523i)11-s + (−1.20 − 1.01i)13-s + (−0.345 − 0.0627i)15-s + (1.56 + 0.904i)17-s + (0.530 − 0.306i)19-s + (−0.465 + 1.25i)21-s + (0.156 − 0.886i)23-s + (0.671 − 0.563i)25-s + (−0.999 − 0.0147i)27-s + (0.574 + 0.685i)29-s + (−0.496 − 0.0874i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.253443 - 0.869879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.253443 - 0.869879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.873 + 1.49i)T \) |
| good | 5 | \( 1 + (0.268 + 0.738i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (3.49 - 0.616i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (4.77 + 1.73i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (4.34 + 3.64i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-6.45 - 3.72i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.31 + 1.33i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.749 + 4.25i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.09 - 3.68i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.76 + 0.487i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.526 + 0.911i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.18 - 1.41i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.64 + 10.0i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.36 - 7.73i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 3.08iT - 53T^{2} \) |
| 59 | \( 1 + (-5.15 + 1.87i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.62 + 9.24i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (6.27 - 7.47i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.72 - 2.99i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (4.99 + 8.64i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.929 + 1.10i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.12 + 0.946i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (3.10 - 1.79i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.87 + 1.04i)T + (74.3 + 62.3i)T^{2} \) |
| 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
−10.54682815588003346019695070560, −9.931057780600117529737582683304, −8.805408272801240579071180976651, −7.969878013440425473983258513855, −7.24532086521998133934826948548, −6.05566483775546159095814441891, −5.22458366925431692111186378187, −3.27161855141507245817459836886, −2.68106695304875925903791134217, −0.51126395811254669430777684397,
2.65336797862382717653063782686, 3.35765477697567888740579161951, 4.72217794301311249298746911272, 5.62226737462969182257826322510, 7.21926865480274963380184138010, 7.63606469126417610938436283008, 9.142453980097362104462048165736, 9.990764445727498606741284459314, 10.07164242899825266870054801150, 11.44357144016292381707802014559
