| L(s) = 1 | + (−1.35 − 1.61i)2-s + (0.767 − 2.10i)3-s + (2.00 − 11.3i)4-s + (−3.09 − 17.5i)5-s + (−4.45 + 1.62i)6-s + (14.2 − 24.6i)7-s + (−50.4 + 29.1i)8-s + (58.1 + 48.8i)9-s + (−24.2 + 28.8i)10-s + (62.8 + 108. i)11-s + (−22.4 − 12.9i)12-s + (6.89 + 18.9i)13-s + (−59.3 + 10.4i)14-s + (−39.3 − 6.93i)15-s + (−57.6 − 20.9i)16-s + (57.9 − 48.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.339 − 0.404i)2-s + (0.0852 − 0.234i)3-s + (0.125 − 0.709i)4-s + (−0.123 − 0.701i)5-s + (−0.123 + 0.0450i)6-s + (0.290 − 0.503i)7-s + (−0.787 + 0.454i)8-s + (0.718 + 0.602i)9-s + (−0.242 + 0.288i)10-s + (0.519 + 0.899i)11-s + (−0.155 − 0.0898i)12-s + (0.0408 + 0.112i)13-s + (−0.302 + 0.0533i)14-s + (−0.174 − 0.0308i)15-s + (−0.225 − 0.0820i)16-s + (0.200 − 0.168i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0389 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0389 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.801918 - 0.771301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.801918 - 0.771301i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-220. + 285. i)T \) |
| good | 2 | \( 1 + (1.35 + 1.61i)T + (-2.77 + 15.7i)T^{2} \) |
| 3 | \( 1 + (-0.767 + 2.10i)T + (-62.0 - 52.0i)T^{2} \) |
| 5 | \( 1 + (3.09 + 17.5i)T + (-587. + 213. i)T^{2} \) |
| 7 | \( 1 + (-14.2 + 24.6i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-62.8 - 108. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-6.89 - 18.9i)T + (-2.18e4 + 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-57.9 + 48.6i)T + (1.45e4 - 8.22e4i)T^{2} \) |
| 23 | \( 1 + (49.3 - 279. i)T + (-2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (785. - 936. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (117. + 67.8i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 871. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (207. - 568. i)T + (-2.16e6 - 1.81e6i)T^{2} \) |
| 43 | \( 1 + (557. + 3.16e3i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (1.11e3 + 936. i)T + (8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + (4.74e3 + 836. i)T + (7.41e6 + 2.69e6i)T^{2} \) |
| 59 | \( 1 + (-2.44e3 - 2.90e3i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (1.18e3 - 6.74e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (3.01e3 - 3.59e3i)T + (-3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-3.05e3 + 539. i)T + (2.38e7 - 8.69e6i)T^{2} \) |
| 73 | \( 1 + (4.97e3 + 1.81e3i)T + (2.17e7 + 1.82e7i)T^{2} \) |
| 79 | \( 1 + (-3.47e3 + 9.54e3i)T + (-2.98e7 - 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-6.22e3 + 1.07e4i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-730. - 2.00e3i)T + (-4.80e7 + 4.03e7i)T^{2} \) |
| 97 | \( 1 + (-9.93e3 - 1.18e4i)T + (-1.53e7 + 8.71e7i)T^{2} \) |
| 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
−17.64477288780766762825821289295, −16.21008233615462492213683916180, −14.81930289065794752119188587846, −13.39725263760388351094155941832, −11.90228628173770017398032203538, −10.43989932666024118025995468308, −9.138777790959014067451666563597, −7.21311058306325245225204352976, −4.89260604545248703826344946328, −1.44443695272573874379562817005,
3.49852989101164531414095164899, 6.40794830547127278731905831588, 7.964871223374938477777960173614, 9.445711168612944330159551029425, 11.31217718752992605311434423519, 12.60784564496908403362760390239, 14.47918971004619586427996534420, 15.60191329784013224920234575088, 16.70840622885886404183177734240, 18.10050997605262546215385810145
