| L(s) = 1 | + 1.74·2-s + (7.37 − 12.7i)3-s − 28.9·4-s + (−8.29 + 14.3i)5-s + (12.8 − 22.2i)6-s + (−98.7 − 171. i)7-s − 106.·8-s + (12.6 + 21.9i)9-s + (−14.4 + 25.0i)10-s − 376.·11-s + (−213. + 369. i)12-s + (−55.5 − 96.2i)13-s + (−172. − 298. i)14-s + (122. + 212. i)15-s + 741.·16-s + (−786. − 1.36e3i)17-s + ⋯ |
| L(s) = 1 | + 0.308·2-s + (0.473 − 0.819i)3-s − 0.904·4-s + (−0.148 + 0.257i)5-s + (0.145 − 0.252i)6-s + (−0.761 − 1.31i)7-s − 0.587·8-s + (0.0522 + 0.0905i)9-s + (−0.0457 + 0.0792i)10-s − 0.938·11-s + (−0.428 + 0.741i)12-s + (−0.0911 − 0.157i)13-s + (−0.234 − 0.406i)14-s + (0.140 + 0.243i)15-s + 0.724·16-s + (−0.659 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.232480 - 0.923063i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.232480 - 0.923063i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.98e3 + 1.14e4i)T \) |
| good | 2 | \( 1 - 1.74T + 32T^{2} \) |
| 3 | \( 1 + (-7.37 + 12.7i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (8.29 - 14.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (98.7 + 171. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + 376.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (55.5 + 96.2i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (786. + 1.36e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-444. + 769. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-428. + 741. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-12.2 - 21.1i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-2.22e3 + 3.84e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.15e3 - 5.46e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 5.20e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 1.33e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-2.12e3 + 3.68e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + 4.94e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (1.70e4 + 2.94e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.10e4 + 3.64e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (1.94e4 + 3.36e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (1.27e4 + 2.20e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.78e4 - 8.29e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.03e3 - 1.79e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (7.30e4 - 1.26e5i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.89e4T + 8.58e9T^{2} \) |
| 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
−13.85327147232206958349660824521, −13.50858409967213007631408748345, −12.64017619720716390962755685047, −10.74690916954533346047120154607, −9.471413809022138950838475263371, −7.888947050339586659933869448802, −6.88875152960968160232838396741, −4.77747795841552057659401376132, −3.06404873643364958740461839416, −0.44777139808328150145092053179,
3.08157456964692314733226017521, 4.53665374848196007532433372082, 5.93936188653486540196539228417, 8.437542810401349719793447789495, 9.206235404802995205829774150417, 10.26580948379974427649203343163, 12.30086730919340604289909357285, 12.99030867393103136004296009879, 14.46423415403037350463915050062, 15.39000819382786241839860120024
