| L(s) = 1 | + (−2.16 + 0.789i)2-s + (−0.930 + 5.27i)3-s + (−2.04 + 1.71i)4-s + (6.28 + 5.27i)5-s + (−2.14 − 12.1i)6-s + (−1.44 − 2.50i)7-s + (12.3 − 21.3i)8-s + (−1.60 − 0.584i)9-s + (−17.8 − 6.48i)10-s + (14.1 − 24.4i)11-s + (−7.15 − 12.3i)12-s + (14.7 + 83.9i)13-s + (5.11 + 4.29i)14-s + (−33.6 + 28.2i)15-s + (−6.16 + 34.9i)16-s + (33.2 − 12.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.766 + 0.279i)2-s + (−0.179 + 1.01i)3-s + (−0.255 + 0.214i)4-s + (0.562 + 0.471i)5-s + (−0.146 − 0.828i)6-s + (−0.0780 − 0.135i)7-s + (0.544 − 0.942i)8-s + (−0.0595 − 0.0216i)9-s + (−0.563 − 0.204i)10-s + (0.386 − 0.670i)11-s + (−0.172 − 0.298i)12-s + (0.315 + 1.79i)13-s + (0.0976 + 0.0819i)14-s + (−0.579 + 0.486i)15-s + (−0.0962 + 0.546i)16-s + (0.474 − 0.172i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.455683 + 0.546659i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.455683 + 0.546659i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (70.3 + 43.7i)T \) |
| good | 2 | \( 1 + (2.16 - 0.789i)T + (6.12 - 5.14i)T^{2} \) |
| 3 | \( 1 + (0.930 - 5.27i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (-6.28 - 5.27i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (1.44 + 2.50i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-14.1 + 24.4i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-14.7 - 83.9i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-33.2 + 12.1i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-131. + 110. i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (62.1 + 22.6i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (76.2 + 132. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 129.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (27.0 - 153. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-242. - 203. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (450. + 163. i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + (-518. + 435. i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (202. - 73.5i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (164. - 137. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (417. + 151. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (-178. - 150. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + (140. - 797. i)T + (-3.65e5 - 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-17.1 + 97.0i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-143. - 247. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (71.1 + 403. i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (841. - 306. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−18.34007796331822890017989942544, −16.74994659925596568291311888702, −16.44883049497617392108099474425, −14.69568412224115031402209514158, −13.31675854661902315429674890537, −11.13320142501329000227629879898, −9.840495679868420710053930346140, −8.854568839471195026996025542152, −6.72037316584197236652191422203, −4.23037200200293134621166627270,
1.34096291675533610155543614125, 5.60801546935119180512152660393, 7.68003656063210878753728929118, 9.220589268558198789323348258194, 10.57449457135910202073950431971, 12.49375242268468069865667237678, 13.36712233578494372713968868810, 15.03520685615536487723264193056, 17.15503601222617151150726521735, 17.76000028412176176628328231273
