| L(s) = 1 | + (0.244 + 1.39i)2-s + (1.00 + 0.143i)3-s + (−1.88 + 0.681i)4-s + (−0.573 + 2.16i)5-s + (0.0445 + 1.42i)6-s + (−1.31 + 1.13i)7-s + (−1.41 − 2.45i)8-s + (−1.89 − 0.557i)9-s + (−3.15 − 0.269i)10-s + (2.12 − 3.31i)11-s + (−1.97 + 0.411i)12-s + (−3.71 + 4.29i)13-s + (−1.90 − 1.55i)14-s + (−0.884 + 2.08i)15-s + (3.06 − 2.56i)16-s + (2.13 + 4.67i)17-s + ⋯ |
| L(s) = 1 | + (0.173 + 0.984i)2-s + (0.577 + 0.0830i)3-s + (−0.940 + 0.340i)4-s + (−0.256 + 0.966i)5-s + (0.0181 + 0.583i)6-s + (−0.497 + 0.430i)7-s + (−0.498 − 0.866i)8-s + (−0.632 − 0.185i)9-s + (−0.996 − 0.0852i)10-s + (0.641 − 0.998i)11-s + (−0.571 + 0.118i)12-s + (−1.03 + 1.19i)13-s + (−0.510 − 0.414i)14-s + (−0.228 + 0.537i)15-s + (0.767 − 0.641i)16-s + (0.517 + 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.317599 - 0.598124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.317599 - 0.598124i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.244 - 1.39i)T \) |
| 5 | \( 1 + (0.573 - 2.16i)T \) |
| 23 | \( 1 + (-0.315 + 4.78i)T \) |
| good | 3 | \( 1 + (-1.00 - 0.143i)T + (2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (1.31 - 1.13i)T + (0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-2.12 + 3.31i)T + (-4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (3.71 - 4.29i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.13 - 4.67i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (2.68 + 1.22i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (3.48 - 1.59i)T + (18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (4.40 - 0.633i)T + (29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-2.76 + 9.42i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.371 + 0.109i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.44 - 10.0i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + (4.29 - 3.71i)T + (7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-1.53 + 1.76i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.784 - 5.45i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (8.13 - 5.22i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (0.510 + 0.794i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (2.99 + 1.36i)T + (47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.412 + 0.476i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-5.71 - 1.67i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (17.3 + 2.49i)T + (85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-16.5 + 4.86i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.51470087058866187813312715995, −9.303540175417798059882238249125, −9.000140479946384805912562203520, −8.052220272925645973125892737087, −7.19555595114415425588216536688, −6.27895871346298569560703058081, −5.86130692831704153897748585359, −4.27305142439603520381176518820, −3.51355123159350545308913892618, −2.52931295892007908408743056237,
0.26897606413780196801544717648, 1.81979630942414456875768464307, 3.00689126003094566480382799356, 3.89309583702830762557235662195, 4.95559066942783169074442504586, 5.61956208401927979811709877044, 7.29895070562637087445924599967, 7.992366259987039381106568245835, 8.957963822203468408386776373594, 9.634239789753329711100320050699
