| L(s) = 1 | + (2.98 + 2.50i)2-s + (−3.72 + 1.35i)3-s + (1.25 + 7.09i)4-s + (2.58 − 14.6i)5-s + (−14.5 − 5.28i)6-s + (5.35 + 9.26i)7-s + (1.55 − 2.68i)8-s + (−8.65 + 7.25i)9-s + (44.5 − 37.3i)10-s + (−14.6 + 25.3i)11-s + (−14.2 − 24.7i)12-s + (−76.4 − 27.8i)13-s + (−7.24 + 41.0i)14-s + (10.2 + 58.1i)15-s + (65.5 − 23.8i)16-s + (61.3 + 51.4i)17-s + ⋯ |
| L(s) = 1 | + (1.05 + 0.886i)2-s + (−0.716 + 0.260i)3-s + (0.156 + 0.886i)4-s + (0.231 − 1.31i)5-s + (−0.988 − 0.359i)6-s + (0.288 + 0.500i)7-s + (0.0686 − 0.118i)8-s + (−0.320 + 0.268i)9-s + (1.40 − 1.18i)10-s + (−0.400 + 0.694i)11-s + (−0.343 − 0.594i)12-s + (−1.63 − 0.593i)13-s + (−0.138 + 0.784i)14-s + (0.176 + 1.00i)15-s + (1.02 − 0.372i)16-s + (0.875 + 0.734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.635 - 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.28278 + 0.605611i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28278 + 0.605611i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-81.6 + 13.9i)T \) |
| good | 2 | \( 1 + (-2.98 - 2.50i)T + (1.38 + 7.87i)T^{2} \) |
| 3 | \( 1 + (3.72 - 1.35i)T + (20.6 - 17.3i)T^{2} \) |
| 5 | \( 1 + (-2.58 + 14.6i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (-5.35 - 9.26i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (14.6 - 25.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (76.4 + 27.8i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-61.3 - 51.4i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-11.6 - 66.1i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-1.78 + 1.49i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (68.9 + 119. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 305.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-213. + 77.8i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (29.9 - 169. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (324. - 271. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (28.7 + 163. i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-355. - 298. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (106. + 603. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-166. + 139. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (9.51 - 53.9i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-130. + 47.5i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (436. - 158. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-43.2 - 74.9i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (443. + 161. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (326. + 274. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−17.50957168605153546237843705775, −16.74227046724546768563905782664, −15.65559748828723925529755893317, −14.46989990067033071365975545149, −12.90512838208524406267470363636, −12.08181958564353123064620529743, −9.860529537245499824275815959329, −7.76122188371221757832566840807, −5.46242042914934957657555255337, −4.99145201442911442842822967994,
3.00514936460027544850512638055, 5.32963370730844641234529717070, 7.15453061456075717790826901700, 10.26731175785302080651057257117, 11.35883341886865030819417442986, 12.24264207395057901899771343049, 14.02188932938363940778153300763, 14.51319666273638491321037655863, 16.75946964908708131000956605644, 18.00466338729526964985198957725
