| L(s) = 1 | + (−1.30 − 0.556i)2-s + (−1.80 + 0.910i)3-s + (1.38 + 1.44i)4-s + (−1.65 − 3.73i)5-s + (2.85 − 0.177i)6-s + (2.41 + 1.44i)7-s + (−0.991 − 2.64i)8-s + (0.653 − 0.880i)9-s + (0.0751 + 5.77i)10-s + (0.243 + 0.190i)11-s + (−3.81 − 1.35i)12-s + (1.10 − 1.05i)13-s + (−2.33 − 3.22i)14-s + (6.38 + 5.24i)15-s + (−0.184 + 3.99i)16-s + (−1.20 − 1.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.919 − 0.393i)2-s + (−1.04 + 0.525i)3-s + (0.690 + 0.723i)4-s + (−0.739 − 1.66i)5-s + (1.16 − 0.0725i)6-s + (0.911 + 0.546i)7-s + (−0.350 − 0.936i)8-s + (0.217 − 0.293i)9-s + (0.0237 + 1.82i)10-s + (0.0735 + 0.0573i)11-s + (−1.10 − 0.392i)12-s + (0.307 − 0.292i)13-s + (−0.623 − 0.861i)14-s + (1.64 + 1.35i)15-s + (−0.0460 + 0.998i)16-s + (−0.292 − 0.356i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0332193 - 0.238512i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0332193 - 0.238512i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 + 0.556i)T \) |
| good | 3 | \( 1 + (1.80 - 0.910i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (1.65 + 3.73i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-2.41 - 1.44i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-0.243 - 0.190i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.10 + 1.05i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.20 + 1.47i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (0.355 + 2.04i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-5.12 - 2.42i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (0.954 + 0.540i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (9.64 - 1.91i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (3.31 + 2.09i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (6.52 + 7.20i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (2.86 + 8.67i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (9.87 + 5.27i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (7.19 - 4.07i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (6.57 - 6.90i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.206 - 2.79i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-1.87 - 1.62i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (-8.38 - 1.24i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (-11.7 + 7.02i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-6.03 - 1.83i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-1.25 + 0.794i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (13.5 - 6.39i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (5.19 + 3.47i)T + (37.1 + 89.6i)T^{2} \) |
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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
−10.79071640868091730197832061911, −9.439724715632430227834856563690, −8.777225902618172451718133431849, −8.159711956558982280794787147810, −7.07231605399150167872164682959, −5.41481291781162799120805893697, −4.98357330507927069624360008308, −3.75532348796701535227176645789, −1.67489725495316245557913264575, −0.22649936205137890061903281529,
1.60972645194887577400740471169, 3.33845062444459320603485923233, 4.97601216263539761300198949115, 6.38812067930001091123075709141, 6.67061328519350270460980862251, 7.59504217829060594585522953990, 8.254978737534528862654867536412, 9.696792995751049612003389498318, 10.88282565729568159934776381648, 11.07760104849646929275173435840
