| L(s) = 1 | + (−1.29 + 0.579i)2-s + (0.597 + 2.03i)3-s + (1.32 − 1.49i)4-s + (−1.54 + 2.40i)5-s + (−1.94 − 2.27i)6-s + (−0.249 − 1.73i)7-s + (−0.847 + 2.69i)8-s + (−1.25 + 0.806i)9-s + (0.599 − 3.99i)10-s + (−0.620 + 1.35i)11-s + (3.83 + 1.80i)12-s + (−0.727 + 5.05i)13-s + (1.32 + 2.09i)14-s + (−5.80 − 1.70i)15-s + (−0.470 − 3.97i)16-s + (4.17 − 3.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.912 + 0.409i)2-s + (0.344 + 1.17i)3-s + (0.664 − 0.747i)4-s + (−0.690 + 1.07i)5-s + (−0.795 − 0.929i)6-s + (−0.0943 − 0.656i)7-s + (−0.299 + 0.954i)8-s + (−0.418 + 0.268i)9-s + (0.189 − 1.26i)10-s + (−0.187 + 0.409i)11-s + (1.10 + 0.522i)12-s + (−0.201 + 1.40i)13-s + (0.355 + 0.560i)14-s + (−1.49 − 0.440i)15-s + (−0.117 − 0.993i)16-s + (1.01 − 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.371671 + 0.564842i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.371671 + 0.564842i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.29 - 0.579i)T \) |
| 23 | \( 1 + (-4.76 + 0.575i)T \) |
| good | 3 | \( 1 + (-0.597 - 2.03i)T + (-2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (1.54 - 2.40i)T + (-2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.249 + 1.73i)T + (-6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (0.620 - 1.35i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.727 - 5.05i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-4.17 + 3.61i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-4.01 + 4.63i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (4.51 + 5.21i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (0.376 - 1.28i)T + (-26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-1.44 - 2.25i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (0.151 + 0.0973i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (3.19 - 0.936i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 0.713iT - 47T^{2} \) |
| 53 | \( 1 + (7.04 - 1.01i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (9.95 + 1.43i)T + (56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-3.77 + 12.8i)T + (-51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (1.87 + 4.10i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (9.80 - 4.47i)T + (46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (3.48 - 4.02i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.274 - 1.90i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-11.1 + 7.16i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (0.565 + 1.92i)T + (-74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (4.90 - 7.63i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−14.75509129131818319069842560395, −13.94134275259853294701835663111, −11.64636660078885916281773697447, −10.92953642565772226027915909224, −9.850620287934558267306234356124, −9.236932284100426323623265629730, −7.53235755593215071199129604199, −6.88322120560703692243302112349, −4.76733977193908491006338369824, −3.15886656884033572209630736201,
1.22580483711375761620590504139, 3.20796711091308429044459808791, 5.68557289030257547513613316641, 7.56056689712165755404905975735, 8.058585753060439281451290669475, 8.999774624009873796776269689379, 10.44824352195829587557608324598, 11.94836156969920890003918364394, 12.53563950388978168895523634928, 13.12706508367529127008121375716
