| L(s) = 1 | + (−1.04 + 0.956i)2-s + (1.63 + 0.557i)3-s + (0.170 − 1.99i)4-s + (−1.60 + 1.05i)5-s + (−2.24 + 0.987i)6-s + (−1.00 − 0.949i)7-s + (1.72 + 2.23i)8-s + (2.37 + 1.82i)9-s + (0.664 − 2.64i)10-s + (0.968 + 1.92i)11-s + (1.39 − 3.17i)12-s + (5.48 − 2.36i)13-s + (1.95 + 0.0264i)14-s + (−3.23 + 0.837i)15-s + (−3.94 − 0.678i)16-s + (2.40 + 2.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.736 + 0.676i)2-s + (0.946 + 0.322i)3-s + (0.0851 − 0.996i)4-s + (−0.719 + 0.473i)5-s + (−0.915 + 0.403i)6-s + (−0.380 − 0.358i)7-s + (0.611 + 0.791i)8-s + (0.792 + 0.609i)9-s + (0.210 − 0.835i)10-s + (0.291 + 0.581i)11-s + (0.401 − 0.915i)12-s + (1.52 − 0.655i)13-s + (0.522 + 0.00707i)14-s + (−0.834 + 0.216i)15-s + (−0.985 − 0.169i)16-s + (0.584 + 0.696i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.835829 + 0.962677i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.835829 + 0.962677i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.04 - 0.956i)T \) |
| 3 | \( 1 + (-1.63 - 0.557i)T \) |
| good | 5 | \( 1 + (1.60 - 1.05i)T + (1.98 - 4.59i)T^{2} \) |
| 7 | \( 1 + (1.00 + 0.949i)T + (0.407 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-0.968 - 1.92i)T + (-6.56 + 8.82i)T^{2} \) |
| 13 | \( 1 + (-5.48 + 2.36i)T + (8.92 - 9.45i)T^{2} \) |
| 17 | \( 1 + (-2.40 - 2.87i)T + (-2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (0.268 + 0.225i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (0.552 + 0.585i)T + (-1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (6.60 - 0.771i)T + (28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (-2.41 - 10.2i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (7.36 - 1.29i)T + (34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-5.62 - 4.19i)T + (11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (-0.0630 + 1.08i)T + (-42.7 - 4.99i)T^{2} \) |
| 47 | \( 1 + (-3.10 - 0.734i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 + (-4.63 - 8.03i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.43 + 4.84i)T + (-35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-14.0 + 4.20i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (7.42 + 0.867i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (3.22 + 1.17i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.68 + 2.79i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (4.95 - 3.68i)T + (22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (6.36 - 4.74i)T + (23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (0.850 + 2.33i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.829 + 0.545i)T + (38.4 + 89.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.52541823252119018538584961014, −9.871628981395133484388779431159, −8.828895301171275292526018268675, −8.252964104108425930838812009702, −7.40029871389529708215683339177, −6.72481900267637710502582401673, −5.49672015962053213351422451888, −4.07148678938094839115724628584, −3.26884027482010620990667893465, −1.50539590602636961020086525750,
0.893997965064520733532850876990, 2.29680589309063197121857288655, 3.60606425813899255125579560239, 4.04864629467212729212324419199, 6.01630763058076689092387473862, 7.18642893363281738080880827058, 7.989656291707159181996359460945, 8.754013909301636923942768846622, 9.179841772015934576218063344062, 10.12275503049156442354777901173
