| 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 | \( 1 + (2.87 - 1.04i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.44357144016292381707802014559, −10.07164242899825266870054801150, −9.990764445727498606741284459314, −9.142453980097362104462048165736, −7.63606469126417610938436283008, −7.21926865480274963380184138010, −5.62226737462969182257826322510, −4.72217794301311249298746911272, −3.35765477697567888740579161951, −2.65336797862382717653063782686,
0.51126395811254669430777684397, 2.68106695304875925903791134217, 3.27161855141507245817459836886, 5.22458366925431692111186378187, 6.05566483775546159095814441891, 7.24532086521998133934826948548, 7.969878013440425473983258513855, 8.805408272801240579071180976651, 9.931057780600117529737582683304, 10.54682815588003346019695070560
