| L(s) = 1 | + (0.315 + 0.229i)2-s + (−3.41 − 3.91i)3-s + (−2.42 − 7.46i)4-s + (−2.50 − 3.44i)5-s + (−0.179 − 2.01i)6-s + (11.0 − 3.60i)7-s + (1.90 − 5.87i)8-s + (−3.68 + 26.7i)9-s − 1.65i·10-s + (25.5 − 26.0i)11-s + (−20.9 + 34.9i)12-s + (−19.1 + 26.3i)13-s + (4.32 + 1.40i)14-s + (−4.94 + 21.5i)15-s + (−48.8 + 35.4i)16-s + (76.8 − 55.8i)17-s + ⋯ |
| L(s) = 1 | + (0.111 + 0.0809i)2-s + (−0.657 − 0.753i)3-s + (−0.303 − 0.933i)4-s + (−0.223 − 0.308i)5-s + (−0.0122 − 0.137i)6-s + (0.598 − 0.194i)7-s + (0.0843 − 0.259i)8-s + (−0.136 + 0.990i)9-s − 0.0524i·10-s + (0.699 − 0.714i)11-s + (−0.504 + 0.841i)12-s + (−0.408 + 0.562i)13-s + (0.0825 + 0.0268i)14-s + (−0.0851 + 0.371i)15-s + (−0.763 + 0.554i)16-s + (1.09 − 0.796i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.658280 - 0.747673i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.658280 - 0.747673i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.41 + 3.91i)T \) |
| 11 | \( 1 + (-25.5 + 26.0i)T \) |
| good | 2 | \( 1 + (-0.315 - 0.229i)T + (2.47 + 7.60i)T^{2} \) |
| 5 | \( 1 + (2.50 + 3.44i)T + (-38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (-11.0 + 3.60i)T + (277. - 201. i)T^{2} \) |
| 13 | \( 1 + (19.1 - 26.3i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-76.8 + 55.8i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-55.1 - 17.9i)T + (5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 78.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (49.8 + 153. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-35.3 - 25.7i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (27.2 + 83.8i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (146. - 451. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 - 285. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-525. - 170. i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (165. - 227. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (70.5 - 22.9i)T + (1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (480. + 661. i)T + (-7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 376.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (133. + 184. i)T + (-1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-478. + 155. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (414. - 570. i)T + (-1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-850. + 617. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 - 661. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-982. - 714. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
| 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
−16.07435216004465936410825875220, −14.36935255754685431234053363217, −13.71742552258936596185282196621, −12.07076554295905890994238441736, −11.15122684633408367343326742325, −9.564258438426854417002673162681, −7.78304425171573335374226746225, −6.17351420328207124332346272748, −4.84381467171546545812700747909, −1.09600953559826601336616749738,
3.64356374244648980901189413987, 5.16272425526080560412414216102, 7.28497257042255858156709309030, 8.888840190894802517643510862025, 10.39320075408703179483289415585, 11.76803025708845497167679078954, 12.50102694906388812274733605952, 14.38608840991320884925397604438, 15.35632019609976501213022918609, 16.81545444443189559701782249709
