| L(s) = 1 | + (−1.96 − 0.465i)2-s + (1.31 + 1.12i)3-s + (1.85 + 0.932i)4-s + (−1.65 + 2.22i)5-s + (−2.07 − 2.81i)6-s + (0.510 + 0.335i)7-s + (−0.120 − 0.101i)8-s + (0.483 + 2.96i)9-s + (4.28 − 3.59i)10-s + (3.71 − 0.434i)11-s + (1.40 + 3.31i)12-s + (−0.980 + 3.27i)13-s + (−0.847 − 0.897i)14-s + (−4.67 + 1.07i)15-s + (−2.29 − 3.07i)16-s + (−1.31 − 7.45i)17-s + ⋯ |
| L(s) = 1 | + (−1.38 − 0.329i)2-s + (0.761 + 0.647i)3-s + (0.928 + 0.466i)4-s + (−0.739 + 0.992i)5-s + (−0.845 − 1.15i)6-s + (0.193 + 0.126i)7-s + (−0.0426 − 0.0357i)8-s + (0.161 + 0.986i)9-s + (1.35 − 1.13i)10-s + (1.12 − 0.131i)11-s + (0.405 + 0.956i)12-s + (−0.272 + 0.908i)13-s + (−0.226 − 0.239i)14-s + (−1.20 + 0.277i)15-s + (−0.572 − 0.769i)16-s + (−0.318 − 1.80i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.520002 + 0.265481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.520002 + 0.265481i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.31 - 1.12i)T \) |
| good | 2 | \( 1 + (1.96 + 0.465i)T + (1.78 + 0.897i)T^{2} \) |
| 5 | \( 1 + (1.65 - 2.22i)T + (-1.43 - 4.78i)T^{2} \) |
| 7 | \( 1 + (-0.510 - 0.335i)T + (2.77 + 6.42i)T^{2} \) |
| 11 | \( 1 + (-3.71 + 0.434i)T + (10.7 - 2.53i)T^{2} \) |
| 13 | \( 1 + (0.980 - 3.27i)T + (-10.8 - 7.14i)T^{2} \) |
| 17 | \( 1 + (1.31 + 7.45i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.132 + 0.751i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (1.81 - 1.19i)T + (9.10 - 21.1i)T^{2} \) |
| 29 | \( 1 + (-3.11 + 3.30i)T + (-1.68 - 28.9i)T^{2} \) |
| 31 | \( 1 + (0.475 + 8.17i)T + (-30.7 + 3.59i)T^{2} \) |
| 37 | \( 1 + (-3.96 - 1.44i)T + (28.3 + 23.7i)T^{2} \) |
| 41 | \( 1 + (-8.79 + 2.08i)T + (36.6 - 18.4i)T^{2} \) |
| 43 | \( 1 + (-0.695 + 1.61i)T + (-29.5 - 31.2i)T^{2} \) |
| 47 | \( 1 + (0.684 - 11.7i)T + (-46.6 - 5.45i)T^{2} \) |
| 53 | \( 1 + (2.80 + 4.85i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.50 + 0.175i)T + (57.4 + 13.6i)T^{2} \) |
| 61 | \( 1 + (-2.70 + 1.35i)T + (36.4 - 48.9i)T^{2} \) |
| 67 | \( 1 + (-5.25 - 5.56i)T + (-3.89 + 66.8i)T^{2} \) |
| 71 | \( 1 + (5.19 - 4.36i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.438 - 0.367i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.96 - 2.12i)T + (70.5 + 35.4i)T^{2} \) |
| 83 | \( 1 + (6.02 + 1.42i)T + (74.1 + 37.2i)T^{2} \) |
| 89 | \( 1 + (3.52 + 2.95i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (6.28 + 8.43i)T + (-27.8 + 92.9i)T^{2} \) |
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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
−14.56036427587668597721950403233, −13.89517101983312275302870394869, −11.51808514539141951635999993495, −11.25890861465873407417358683582, −9.737688213635119446298330461343, −9.214778695400270678577415826739, −7.927561896512668972642100152073, −7.00984702531657584129403240969, −4.30017821116686214027184277424, −2.57599475822761414690788062231,
1.28430773702086755205563708449, 4.06561358243036604711518666988, 6.52382323851625836908032472145, 7.84087650506654134802241338776, 8.424450920501728273486158391940, 9.236610074603381359486583379501, 10.58775676223632723122003992544, 12.20857185280012260321199045468, 12.89792887069029189985537221776, 14.44048691000316373112421632617
