| L(s) = 1 | + (−1.14 − 0.415i)2-s + (−3.98 + 15.0i)3-s + (−23.3 − 19.6i)4-s + (10.2 − 58.3i)5-s + (10.8 − 15.5i)6-s + (144. − 121. i)7-s + (37.9 + 65.7i)8-s + (−211. − 120. i)9-s + (−36.0 + 62.3i)10-s + (−82.8 − 469. i)11-s + (388. − 274. i)12-s + (−973. + 354. i)13-s + (−215. + 78.2i)14-s + (838. + 387. i)15-s + (153. + 871. i)16-s + (763. − 1.32e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.201 − 0.0734i)2-s + (−0.255 + 0.966i)3-s + (−0.730 − 0.613i)4-s + (0.184 − 1.04i)5-s + (0.122 − 0.176i)6-s + (1.11 − 0.934i)7-s + (0.209 + 0.363i)8-s + (−0.869 − 0.493i)9-s + (−0.113 + 0.197i)10-s + (−0.206 − 1.17i)11-s + (0.779 − 0.549i)12-s + (−1.59 + 0.581i)13-s + (−0.293 + 0.106i)14-s + (0.962 + 0.444i)15-s + (0.150 + 0.850i)16-s + (0.640 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0185 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0185 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.637590 - 0.649534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.637590 - 0.649534i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.98 - 15.0i)T \) |
| good | 2 | \( 1 + (1.14 + 0.415i)T + (24.5 + 20.5i)T^{2} \) |
| 5 | \( 1 + (-10.2 + 58.3i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-144. + 121. i)T + (2.91e3 - 1.65e4i)T^{2} \) |
| 11 | \( 1 + (82.8 + 469. i)T + (-1.51e5 + 5.50e4i)T^{2} \) |
| 13 | \( 1 + (973. - 354. i)T + (2.84e5 - 2.38e5i)T^{2} \) |
| 17 | \( 1 + (-763. + 1.32e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-583. - 1.01e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.34e3 + 1.97e3i)T + (1.11e6 + 6.33e6i)T^{2} \) |
| 29 | \( 1 + (-4.05e3 - 1.47e3i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (422. + 354. i)T + (4.97e6 + 2.81e7i)T^{2} \) |
| 37 | \( 1 + (402. - 697. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-1.03e3 + 375. i)T + (8.87e7 - 7.44e7i)T^{2} \) |
| 43 | \( 1 + (-569. - 3.22e3i)T + (-1.38e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (7.10e3 - 5.96e3i)T + (3.98e7 - 2.25e8i)T^{2} \) |
| 53 | \( 1 - 3.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (3.36e3 - 1.91e4i)T + (-6.71e8 - 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-5.47e3 + 4.59e3i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.50e4 + 9.12e3i)T + (1.03e9 - 8.67e8i)T^{2} \) |
| 71 | \( 1 + (-3.18e4 + 5.51e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (3.44e3 + 5.96e3i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-2.81e4 - 1.02e4i)T + (2.35e9 + 1.97e9i)T^{2} \) |
| 83 | \( 1 + (3.08e4 + 1.12e4i)T + (3.01e9 + 2.53e9i)T^{2} \) |
| 89 | \( 1 + (3.36e3 + 5.82e3i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (-7.58e3 - 4.29e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
| 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
−16.38519204713058708275893705510, −14.46163087827627970374184415964, −13.94448968142704222954728661937, −11.91802213928202338753394124238, −10.51000408880096685369697876290, −9.472088841259191337475718505284, −8.214514154075847759287513512745, −5.30270941971639690341786260570, −4.50864343272878050757956181679, −0.69042920058130058324657848505,
2.38185374084157419049115746647, 5.21182959438938097591882214506, 7.24643727198600791036484780071, 8.169472343373254975301844480685, 10.04052631788169453068502030764, 11.83682945949157583646195042378, 12.63492550853546262970814917194, 14.20026153848671487742372074974, 15.04822144227774267565112581131, 17.36779927974895424715331292224
