| L(s) = 1 | + (−0.104 − 0.994i)2-s + (0.525 + 0.583i)3-s + (−0.978 + 0.207i)4-s + (−1.47 − 1.68i)5-s + (0.525 − 0.583i)6-s − 3.06·7-s + (0.309 + 0.951i)8-s + (0.249 − 2.37i)9-s + (−1.52 + 1.63i)10-s + (3.96 − 2.87i)11-s + (−0.635 − 0.461i)12-s + (−0.491 + 4.67i)13-s + (0.320 + 3.05i)14-s + (0.209 − 1.74i)15-s + (0.913 − 0.406i)16-s + (−3.48 − 0.741i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 − 0.703i)2-s + (0.303 + 0.336i)3-s + (−0.489 + 0.103i)4-s + (−0.657 − 0.753i)5-s + (0.214 − 0.238i)6-s − 1.15·7-s + (0.109 + 0.336i)8-s + (0.0830 − 0.790i)9-s + (−0.481 + 0.518i)10-s + (1.19 − 0.868i)11-s + (−0.183 − 0.133i)12-s + (−0.136 + 1.29i)13-s + (0.0857 + 0.815i)14-s + (0.0541 − 0.449i)15-s + (0.228 − 0.101i)16-s + (−0.845 − 0.179i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0368552 + 0.0727513i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0368552 + 0.0727513i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 + 0.994i)T \) |
| 5 | \( 1 + (1.47 + 1.68i)T \) |
| 19 | \( 1 + (0.409 - 4.33i)T \) |
| good | 3 | \( 1 + (-0.525 - 0.583i)T + (-0.313 + 2.98i)T^{2} \) |
| 7 | \( 1 + 3.06T + 7T^{2} \) |
| 11 | \( 1 + (-3.96 + 2.87i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.491 - 4.67i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (3.48 + 0.741i)T + (15.5 + 6.91i)T^{2} \) |
| 23 | \( 1 + (7.83 + 3.48i)T + (15.3 + 17.0i)T^{2} \) |
| 29 | \( 1 + (2.20 - 0.469i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (-2.08 - 6.42i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.29 + 2.39i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.70 - 2.98i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (4.69 - 8.13i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (10.5 - 2.24i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-4.66 + 0.991i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-9.27 + 4.12i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (1.38 + 0.617i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-6.71 + 7.45i)T + (-7.00 - 66.6i)T^{2} \) |
| 71 | \( 1 + (11.0 + 12.2i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.956 - 9.09i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-1.25 - 1.39i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (3.33 + 10.2i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (3.58 + 1.59i)T + (59.5 + 66.1i)T^{2} \) |
| 97 | \( 1 + (-1.52 - 1.68i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−9.515037980418452268169716710450, −8.818393639773665061813812491913, −8.316764799933804819991071341776, −6.72637684909910484525138290102, −6.23558921759314136071512404748, −4.62158928121664497956688166207, −3.82712537589114754405789257465, −3.35241057306726383824498572602, −1.61014767977280922257934379810, −0.03699376780614702763980836101,
2.21209466746660261296650648181, 3.44647729613291840631399293620, 4.30571423678800399458093058147, 5.60544352578294035976389411616, 6.73224564586905733188976387052, 7.01731846727273202204219054316, 7.948134729250089251931303641538, 8.683854153543800848265187859440, 9.863095840908933792332695579111, 10.21552478889561464815737463487
