| L(s) = 1 | + (−1.32 − 0.495i)2-s + (1.37 − 3.01i)3-s + (1.50 + 1.31i)4-s + (1.26 + 1.84i)5-s + (−3.32 + 3.31i)6-s + (−0.696 − 1.08i)7-s + (−1.34 − 2.48i)8-s + (−5.24 − 6.05i)9-s + (−0.757 − 3.07i)10-s + (−5.89 − 0.847i)11-s + (6.04 − 2.74i)12-s + (0.377 + 0.242i)13-s + (0.385 + 1.78i)14-s + (7.31 − 1.26i)15-s + (0.549 + 3.96i)16-s + (−0.216 + 0.736i)17-s + ⋯ |
| L(s) = 1 | + (−0.936 − 0.350i)2-s + (0.796 − 1.74i)3-s + (0.754 + 0.656i)4-s + (0.564 + 0.825i)5-s + (−1.35 + 1.35i)6-s + (−0.263 − 0.409i)7-s + (−0.475 − 0.879i)8-s + (−1.74 − 2.01i)9-s + (−0.239 − 0.970i)10-s + (−1.77 − 0.255i)11-s + (1.74 − 0.791i)12-s + (0.104 + 0.0672i)13-s + (0.102 + 0.476i)14-s + (1.88 − 0.327i)15-s + (0.137 + 0.990i)16-s + (−0.0524 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.145824 + 0.760058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.145824 + 0.760058i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 + 0.495i)T \) |
| 5 | \( 1 + (-1.26 - 1.84i)T \) |
| 23 | \( 1 + (3.10 + 3.65i)T \) |
| good | 3 | \( 1 + (-1.37 + 3.01i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (0.696 + 1.08i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (5.89 + 0.847i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.377 - 0.242i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.216 - 0.736i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (0.684 + 2.33i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.814 + 2.77i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.51 - 5.51i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (3.49 + 4.03i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (1.47 - 1.70i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.28 + 7.19i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 3.81iT - 47T^{2} \) |
| 53 | \( 1 + (4.54 - 2.92i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.98 + 4.63i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (2.22 - 1.01i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.783 + 5.45i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.97 + 13.7i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.04 - 10.3i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-10.9 - 7.05i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.08 - 4.71i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (5.35 - 11.7i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (8.69 + 7.53i)T + (13.8 + 96.0i)T^{2} \) |
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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
−9.546767515689011504101731206068, −8.489053736636399320292488142981, −8.017541716773961083559709630879, −7.13594596506742946293677065705, −6.69745238481366848805835118623, −5.75562363694837613937943818712, −3.41111441699305000363935191551, −2.60233685683250999150883328359, −1.98303562398201299748364523132, −0.39980908664366242631767676597,
2.16215134150602233773752578131, 3.03097604858427713355605226262, 4.54581312122212103795738536705, 5.33821996851800411870993345345, 5.93937160074939790558840824053, 7.79592523674100116372278749957, 8.208112266451633157781017822086, 9.068068956782122955217434037823, 9.638858156459888857321000719702, 10.20711286150821193672435077815
