| L(s) = 1 | + (−0.338 − 0.930i)2-s + (−13.3 − 2.36i)3-s + (11.5 − 9.65i)4-s + (−30.8 − 25.8i)5-s + (2.33 + 13.2i)6-s + (23.4 + 40.6i)7-s + (−26.6 − 15.3i)8-s + (97.4 + 35.4i)9-s + (−13.6 + 37.4i)10-s + (80.5 − 139. i)11-s + (−176. + 102. i)12-s + (−10.3 + 1.83i)13-s + (29.9 − 35.6i)14-s + (351. + 418. i)15-s + (36.4 − 206. i)16-s + (−103. + 37.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.0846 − 0.232i)2-s + (−1.48 − 0.262i)3-s + (0.719 − 0.603i)4-s + (−1.23 − 1.03i)5-s + (0.0649 + 0.368i)6-s + (0.479 + 0.830i)7-s + (−0.415 − 0.239i)8-s + (1.20 + 0.437i)9-s + (−0.136 + 0.374i)10-s + (0.665 − 1.15i)11-s + (−1.22 + 0.708i)12-s + (−0.0614 + 0.0108i)13-s + (0.152 − 0.181i)14-s + (1.56 + 1.86i)15-s + (0.142 − 0.807i)16-s + (−0.357 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.250239 - 0.586614i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.250239 - 0.586614i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (273. + 235. i)T \) |
| good | 2 | \( 1 + (0.338 + 0.930i)T + (-12.2 + 10.2i)T^{2} \) |
| 3 | \( 1 + (13.3 + 2.36i)T + (76.1 + 27.7i)T^{2} \) |
| 5 | \( 1 + (30.8 + 25.8i)T + (108. + 615. i)T^{2} \) |
| 7 | \( 1 + (-23.4 - 40.6i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-80.5 + 139. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (10.3 - 1.83i)T + (2.68e4 - 9.76e3i)T^{2} \) |
| 17 | \( 1 + (103. - 37.5i)T + (6.39e4 - 5.36e4i)T^{2} \) |
| 23 | \( 1 + (-506. + 424. i)T + (4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (188. - 518. i)T + (-5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (-821. + 474. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.18e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (64.1 + 11.3i)T + (2.65e6 + 9.66e5i)T^{2} \) |
| 43 | \( 1 + (-217. - 182. i)T + (5.93e5 + 3.36e6i)T^{2} \) |
| 47 | \( 1 + (-680. - 247. i)T + (3.73e6 + 3.13e6i)T^{2} \) |
| 53 | \( 1 + (2.49e3 + 2.97e3i)T + (-1.37e6 + 7.77e6i)T^{2} \) |
| 59 | \( 1 + (-939. - 2.58e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (-3.74e3 + 3.14e3i)T + (2.40e6 - 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-1.58e3 + 4.34e3i)T + (-1.54e7 - 1.29e7i)T^{2} \) |
| 71 | \( 1 + (2.41e3 - 2.88e3i)T + (-4.41e6 - 2.50e7i)T^{2} \) |
| 73 | \( 1 + (283. - 1.60e3i)T + (-2.66e7 - 9.71e6i)T^{2} \) |
| 79 | \( 1 + (-1.03e4 - 1.83e3i)T + (3.66e7 + 1.33e7i)T^{2} \) |
| 83 | \( 1 + (-1.33e3 - 2.31e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (660. - 116. i)T + (5.89e7 - 2.14e7i)T^{2} \) |
| 97 | \( 1 + (4.11e3 + 1.12e4i)T + (-6.78e7 + 5.69e7i)T^{2} \) |
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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
−17.04418877955187211062966818543, −16.17408015812116878033335749893, −15.12205747355287301024452129213, −12.57113050473427363690255746538, −11.60162692815129209695674225969, −11.08790673734711870630324319262, −8.643691807965243595937913231418, −6.48879406738333906564508799998, −5.02411475864727691034841267623, −0.73134389226624472353781958121,
4.15064573969368952265953667573, 6.66140446917530472598091880572, 7.53817037148865803235031575186, 10.60863813372296708453358541802, 11.39671948790673150619890667553, 12.25703990781852799937234834992, 14.84876311891346132821607382885, 15.80182660278590727422913590749, 17.05571380086640759918784487846, 17.68520485828010941817002150836
