| L(s) = 1 | + (−1.31 + 0.525i)2-s + (1.78 − 1.24i)3-s + (1.44 − 1.38i)4-s + (−1.31 − 1.80i)5-s + (−1.68 + 2.57i)6-s + (−4.05 − 1.08i)7-s + (−1.17 + 2.57i)8-s + (0.595 − 1.63i)9-s + (2.67 + 1.67i)10-s + (0.636 + 0.367i)11-s + (0.857 − 4.26i)12-s + (−3.61 − 2.53i)13-s + (5.89 − 0.704i)14-s + (−4.60 − 1.57i)15-s + (0.189 − 3.99i)16-s + (−3.04 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.928 + 0.371i)2-s + (1.02 − 0.721i)3-s + (0.723 − 0.690i)4-s + (−0.589 − 0.807i)5-s + (−0.687 + 1.05i)6-s + (−1.53 − 0.410i)7-s + (−0.415 + 0.909i)8-s + (0.198 − 0.545i)9-s + (0.847 + 0.530i)10-s + (0.191 + 0.110i)11-s + (0.247 − 1.23i)12-s + (−1.00 − 0.702i)13-s + (1.57 − 0.188i)14-s + (−1.18 − 0.407i)15-s + (0.0474 − 0.998i)16-s + (−0.737 − 0.343i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.186029 - 0.566885i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.186029 - 0.566885i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 - 0.525i)T \) |
| 5 | \( 1 + (1.31 + 1.80i)T \) |
| 19 | \( 1 + (2.00 - 3.87i)T \) |
| good | 3 | \( 1 + (-1.78 + 1.24i)T + (1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (4.05 + 1.08i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.636 - 0.367i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.61 + 2.53i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (3.04 + 1.41i)T + (10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-7.79 - 0.682i)T + (22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (-2.65 + 7.28i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.23 - 1.29i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.73 + 4.73i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.21 + 6.91i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.898 + 10.2i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (1.71 - 0.799i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.729 + 8.33i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-5.52 + 2.01i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.21 + 4.37i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.86 + 1.80i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-1.55 + 1.85i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.94 + 4.20i)T + (-24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (0.591 - 3.35i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.616 + 2.30i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (2.79 - 0.492i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.00 - 0.467i)T + (62.3 + 74.3i)T^{2} \) |
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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
−10.70362575290514223096319867887, −9.706273111144207075640243700934, −9.014399068105937389065743117871, −8.249688324139695481187182180373, −7.30634200002726075792635486956, −6.80431014720395537091737499151, −5.31127441906196180787252760542, −3.54267724728866999721453773838, −2.28081551727147519608470485358, −0.44192338082384651105616566934,
2.72201834204752333312779379966, 3.08146140654581727984418291495, 4.29982485111115288201489650921, 6.63887176805471675015131138226, 6.97300640982304906837933526833, 8.422524058060973111050556752654, 9.134902486314382262952774572329, 9.677749576197397871260519647204, 10.56734120730854053272768248736, 11.44440445158164584696813396057
