| L(s) = 1 | + (−0.301 − 0.0531i)2-s + (−9.51 + 11.3i)3-s + (−14.9 − 5.44i)4-s + (−0.554 + 0.201i)5-s + (3.47 − 2.91i)6-s + (33.9 + 58.8i)7-s + (8.46 + 4.88i)8-s + (−23.9 − 136. i)9-s + (0.177 − 0.0313i)10-s + (−45.0 + 78.1i)11-s + (203. − 117. i)12-s + (−53.1 − 63.3i)13-s + (−7.12 − 19.5i)14-s + (2.98 − 8.20i)15-s + (192. + 161. i)16-s + (−47.2 + 268. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0754 − 0.0132i)2-s + (−1.05 + 1.26i)3-s + (−0.934 − 0.340i)4-s + (−0.0221 + 0.00807i)5-s + (0.0965 − 0.0809i)6-s + (0.693 + 1.20i)7-s + (0.132 + 0.0763i)8-s + (−0.296 − 1.67i)9-s + (0.00177 − 0.000313i)10-s + (−0.372 + 0.645i)11-s + (1.41 − 0.817i)12-s + (−0.314 − 0.374i)13-s + (−0.0363 − 0.0998i)14-s + (0.0132 − 0.0364i)15-s + (0.752 + 0.631i)16-s + (−0.163 + 0.928i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.196185 + 0.533764i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.196185 + 0.533764i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (315. + 175. i)T \) |
| good | 2 | \( 1 + (0.301 + 0.0531i)T + (15.0 + 5.47i)T^{2} \) |
| 3 | \( 1 + (9.51 - 11.3i)T + (-14.0 - 79.7i)T^{2} \) |
| 5 | \( 1 + (0.554 - 0.201i)T + (478. - 401. i)T^{2} \) |
| 7 | \( 1 + (-33.9 - 58.8i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (45.0 - 78.1i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (53.1 + 63.3i)T + (-4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (47.2 - 268. i)T + (-7.84e4 - 2.85e4i)T^{2} \) |
| 23 | \( 1 + (-878. - 319. i)T + (2.14e5 + 1.79e5i)T^{2} \) |
| 29 | \( 1 + (-1.27e3 + 225. i)T + (6.64e5 - 2.41e5i)T^{2} \) |
| 31 | \( 1 + (87.0 - 50.2i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.63e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.48e3 - 1.77e3i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-501. + 182. i)T + (2.61e6 - 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-216. - 1.23e3i)T + (-4.58e6 + 1.66e6i)T^{2} \) |
| 53 | \( 1 + (201. - 552. i)T + (-6.04e6 - 5.07e6i)T^{2} \) |
| 59 | \( 1 + (-2.44e3 - 431. i)T + (1.13e7 + 4.14e6i)T^{2} \) |
| 61 | \( 1 + (-2.51e3 - 915. i)T + (1.06e7 + 8.89e6i)T^{2} \) |
| 67 | \( 1 + (3.49e3 - 616. i)T + (1.89e7 - 6.89e6i)T^{2} \) |
| 71 | \( 1 + (-2.40e3 - 6.61e3i)T + (-1.94e7 + 1.63e7i)T^{2} \) |
| 73 | \( 1 + (-730. - 613. i)T + (4.93e6 + 2.79e7i)T^{2} \) |
| 79 | \( 1 + (564. - 672. i)T + (-6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (709. + 1.22e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (6.84e3 + 8.16e3i)T + (-1.08e7 + 6.17e7i)T^{2} \) |
| 97 | \( 1 + (-22.7 - 4.01i)T + (8.31e7 + 3.02e7i)T^{2} \) |
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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
−17.78769957186281804901820525401, −17.28147042224639117284968561668, −15.42761512176455524338437490233, −14.91653977899534563680374943291, −12.72582193064141412135894938355, −11.21915020325052411451479762272, −10.01207262981273155020524186564, −8.744591559677555313099590880938, −5.60045999919838531681834574122, −4.62518853427305515111145843332,
0.64021702222262091001740224750, 4.85563729342993232455322691344, 6.92142487532516876132114962898, 8.237579839914963143452232621791, 10.60313232213098377391446621698, 11.96780693622922413113276269864, 13.25730262994231068415152112483, 14.05879688864085962178153576863, 16.69290748722963058903045258998, 17.33975458681886665528315883107
