| L(s) = 1 | + (−0.642 − 1.25i)2-s + (−0.988 − 0.149i)3-s + (−1.17 + 1.61i)4-s + (−1.15 + 0.455i)5-s + (0.447 + 1.34i)6-s + (1.12 − 2.39i)7-s + (2.79 + 0.439i)8-s + (0.955 + 0.294i)9-s + (1.31 + 1.16i)10-s + (0.254 + 0.824i)11-s + (1.40 − 1.42i)12-s + (0.649 + 0.148i)13-s + (−3.73 + 0.128i)14-s + (1.21 − 0.277i)15-s + (−1.24 − 3.80i)16-s + (0.311 − 0.457i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.890i)2-s + (−0.570 − 0.0860i)3-s + (−0.587 + 0.809i)4-s + (−0.518 + 0.203i)5-s + (0.182 + 0.547i)6-s + (0.423 − 0.905i)7-s + (0.987 + 0.155i)8-s + (0.318 + 0.0982i)9-s + (0.416 + 0.369i)10-s + (0.0766 + 0.248i)11-s + (0.404 − 0.411i)12-s + (0.180 + 0.0411i)13-s + (−0.999 + 0.0343i)14-s + (0.313 − 0.0715i)15-s + (−0.310 − 0.950i)16-s + (0.0756 − 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0677109 - 0.538259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0677109 - 0.538259i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 + 1.25i)T \) |
| 3 | \( 1 + (0.988 + 0.149i)T \) |
| 7 | \( 1 + (-1.12 + 2.39i)T \) |
| good | 5 | \( 1 + (1.15 - 0.455i)T + (3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.254 - 0.824i)T + (-9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.649 - 0.148i)T + (11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.311 + 0.457i)T + (-6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-2.37 + 4.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.16 + 3.16i)T + (-8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (6.93 - 3.33i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (3.93 + 6.81i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.364 + 4.86i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (8.05 + 6.42i)T + (9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-7.27 + 5.79i)T + (9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (5.30 + 4.91i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.664 - 8.86i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (0.00836 - 0.0213i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-6.64 - 0.498i)T + (60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (5.13 - 2.96i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.19 + 2.47i)T + (-44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-6.51 - 7.01i)T + (-5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (4.49 + 2.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.743 + 3.25i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-2.94 + 9.55i)T + (-73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 13.1iT - 97T^{2} \) |
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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
−10.51417537026080498828046270268, −9.623482210805668282556671164124, −8.664274994236526655180956697743, −7.44168877368682276940433298044, −7.20427460668306413329591636813, −5.47891365965586761132160748472, −4.33121721529014587664885251122, −3.58568498238002403029348369352, −1.93445586805504280537586205385, −0.40153561833263391665342065839,
1.55638092978589399276947349110, 3.73332055863360326321064609249, 4.96141438255323059801884351175, 5.69176232516571344982782207701, 6.49008670543799548725940085737, 7.77863714435197302625564346571, 8.210985436291911646886417791583, 9.291047340837061058461642819305, 10.00211201763686568938830713167, 11.13653501311960304290426457795
