| L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (0.751 − 0.867i)5-s + (−0.959 − 0.281i)6-s + (0.841 − 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.965 − 0.620i)10-s + (0.649 − 4.51i)11-s + (−0.142 + 0.989i)12-s + (4.74 + 3.05i)13-s + (−0.654 − 0.755i)14-s + (−0.476 − 1.04i)15-s + (0.841 − 0.540i)16-s + (7.79 + 2.28i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.336 − 0.387i)5-s + (−0.391 − 0.115i)6-s + (0.317 − 0.204i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (−0.305 − 0.196i)10-s + (0.195 − 1.36i)11-s + (−0.0410 + 0.285i)12-s + (1.31 + 0.846i)13-s + (−0.175 − 0.201i)14-s + (−0.123 − 0.269i)15-s + (0.210 − 0.135i)16-s + (1.89 + 0.555i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.908294 - 1.56904i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.908294 - 1.56904i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (3.88 + 2.81i)T \) |
| good | 5 | \( 1 + (-0.751 + 0.867i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.649 + 4.51i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-4.74 - 3.05i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-7.79 - 2.28i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.30 + 0.383i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (9.40 + 2.76i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (3.48 + 7.63i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.519 + 0.599i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (3.71 - 4.28i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (1.37 - 3.01i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + (-7.75 + 4.98i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (9.10 + 5.85i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (0.406 + 0.889i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.784 - 5.45i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.324 - 2.25i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (6.98 - 2.05i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (6.21 + 3.99i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (4.68 + 5.40i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.88 + 10.7i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (5.66 - 6.53i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−9.655919611703730626568583159885, −8.929451566413954683020143373367, −8.229118144684592814605857805018, −7.48532558278649059890277561203, −6.01003728739930980512798364194, −5.63487751144489161978104347212, −3.99074151896052092999194060506, −3.39129152608463805332254759448, −1.85734115826955130701793427480, −0.981269211402891156467852859231,
1.58239403718778737573403909759, 3.18613160287890110162869850506, 4.08609682584647672043324380951, 5.42835218722513479458139878720, 5.71301898706851691170809527920, 7.13058945200056911103972113367, 7.66963767601860647525376825290, 8.645278501290461254654867327235, 9.432801502320373719470079069710, 10.20108522945202843331100585560
