| L(s) = 1 | + (0.995 + 0.0950i)2-s + (−0.690 + 0.723i)3-s + (0.981 + 0.189i)4-s + (−1.41 + 0.727i)5-s + (−0.755 + 0.654i)6-s + (1.71 + 2.01i)7-s + (0.959 + 0.281i)8-s + (−0.0475 − 0.998i)9-s + (−1.47 + 0.590i)10-s + (0.284 + 2.98i)11-s + (−0.814 + 0.580i)12-s + (0.805 + 0.115i)13-s + (1.51 + 2.16i)14-s + (0.447 − 1.52i)15-s + (0.928 + 0.371i)16-s + (−0.425 + 1.22i)17-s + ⋯ |
| L(s) = 1 | + (0.703 + 0.0672i)2-s + (−0.398 + 0.417i)3-s + (0.490 + 0.0946i)4-s + (−0.631 + 0.325i)5-s + (−0.308 + 0.267i)6-s + (0.648 + 0.761i)7-s + (0.339 + 0.0996i)8-s + (−0.0158 − 0.332i)9-s + (−0.466 + 0.186i)10-s + (0.0858 + 0.898i)11-s + (−0.235 + 0.167i)12-s + (0.223 + 0.0321i)13-s + (0.405 + 0.579i)14-s + (0.115 − 0.393i)15-s + (0.232 + 0.0929i)16-s + (−0.103 + 0.297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.951970 + 1.49343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.951970 + 1.49343i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.995 - 0.0950i)T \) |
| 3 | \( 1 + (0.690 - 0.723i)T \) |
| 7 | \( 1 + (-1.71 - 2.01i)T \) |
| 23 | \( 1 + (1.41 - 4.58i)T \) |
| good | 5 | \( 1 + (1.41 - 0.727i)T + (2.90 - 4.07i)T^{2} \) |
| 11 | \( 1 + (-0.284 - 2.98i)T + (-10.8 + 2.08i)T^{2} \) |
| 13 | \( 1 + (-0.805 - 0.115i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.425 - 1.22i)T + (-13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (2.26 + 6.54i)T + (-14.9 + 11.7i)T^{2} \) |
| 29 | \( 1 + (-4.34 - 5.01i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (8.42 - 2.04i)T + (27.5 - 14.2i)T^{2} \) |
| 37 | \( 1 + (1.39 - 0.0663i)T + (36.8 - 3.51i)T^{2} \) |
| 41 | \( 1 + (1.77 - 2.76i)T + (-17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (0.668 + 2.27i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + (-8.34 - 4.81i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.82 - 6.13i)T + (-12.4 + 51.5i)T^{2} \) |
| 59 | \( 1 + (0.798 + 1.99i)T + (-42.7 + 40.7i)T^{2} \) |
| 61 | \( 1 + (-2.53 + 2.42i)T + (2.90 - 60.9i)T^{2} \) |
| 67 | \( 1 + (-1.45 - 1.03i)T + (21.9 + 63.3i)T^{2} \) |
| 71 | \( 1 + (0.801 + 1.75i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.71 + 8.90i)T + (-67.7 - 27.1i)T^{2} \) |
| 79 | \( 1 + (-0.629 + 0.800i)T + (-18.6 - 76.7i)T^{2} \) |
| 83 | \( 1 + (1.11 - 0.715i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-2.47 + 10.1i)T + (-79.1 - 40.7i)T^{2} \) |
| 97 | \( 1 + (-2.15 - 1.38i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.67134311014368174926935487116, −9.406818483591523979380750401561, −8.669922887911179718230469655990, −7.52063110456734682901035710845, −6.88613975163365749983409704173, −5.77664367600760551396246949436, −4.97033993512715931057548177289, −4.22664369026072414734222314573, −3.16469306677173542867911903033, −1.87153338305667501141747032585,
0.68731442015738626815939456918, 2.10478316914907906819630960281, 3.72873191732639967432994787112, 4.27800094693003007605951788388, 5.41306349649390352431294809362, 6.19873853781768330684833733054, 7.15738702111822532426871231639, 8.046939790094690044233699269032, 8.537365402548241594830439116460, 10.12615587018037170783559868858
