| 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 | \( 1 + (0.829 - 0.545i)T + (38.4 - 89.0i)T^{2} \) |
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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
−10.12275503049156442354777901173, −9.179841772015934576218063344062, −8.754013909301636923942768846622, −7.989656291707159181996359460945, −7.18642893363281738080880827058, −6.01630763058076689092387473862, −4.04864629467212729212324419199, −3.60606425813899255125579560239, −2.29680589309063197121857288655, −0.893997965064520733532850876990,
1.50539590602636961020086525750, 3.26884027482010620990667893465, 4.07148678938094839115724628584, 5.49672015962053213351422451888, 6.72481900267637710502582401673, 7.40029871389529708215683339177, 8.252964104108425930838812009702, 8.828895301171275292526018268675, 9.871628981395133484388779431159, 10.52541823252119018538584961014
