| L(s) = 1 | + (8.59 + 1.51i)2-s + (11.8 − 14.1i)3-s + (11.4 + 4.16i)4-s + (148. − 54.0i)5-s + (123. − 103. i)6-s + (77.3 + 133. i)7-s + (−391. − 226. i)8-s + (67.2 + 381. i)9-s + (1.35e3 − 239. i)10-s + (−325. + 563. i)11-s + (195. − 112. i)12-s + (−894. − 1.06e3i)13-s + (461. + 1.26e3i)14-s + (998. − 2.74e3i)15-s + (−3.62e3 − 3.03e3i)16-s + (−1.00e3 + 5.68e3i)17-s + ⋯ |
| L(s) = 1 | + (1.07 + 0.189i)2-s + (0.440 − 0.524i)3-s + (0.179 + 0.0651i)4-s + (1.18 − 0.432i)5-s + (0.572 − 0.480i)6-s + (0.225 + 0.390i)7-s + (−0.764 − 0.441i)8-s + (0.0922 + 0.523i)9-s + (1.35 − 0.239i)10-s + (−0.244 + 0.423i)11-s + (0.112 − 0.0652i)12-s + (−0.407 − 0.485i)13-s + (0.168 + 0.462i)14-s + (0.295 − 0.812i)15-s + (−0.884 − 0.741i)16-s + (−0.204 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.82123 - 0.388683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.82123 - 0.388683i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (6.83e3 + 521. i)T \) |
| good | 2 | \( 1 + (-8.59 - 1.51i)T + (60.1 + 21.8i)T^{2} \) |
| 3 | \( 1 + (-11.8 + 14.1i)T + (-126. - 717. i)T^{2} \) |
| 5 | \( 1 + (-148. + 54.0i)T + (1.19e4 - 1.00e4i)T^{2} \) |
| 7 | \( 1 + (-77.3 - 133. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (325. - 563. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (894. + 1.06e3i)T + (-8.38e5 + 4.75e6i)T^{2} \) |
| 17 | \( 1 + (1.00e3 - 5.68e3i)T + (-2.26e7 - 8.25e6i)T^{2} \) |
| 23 | \( 1 + (5.53e3 + 2.01e3i)T + (1.13e8 + 9.51e7i)T^{2} \) |
| 29 | \( 1 + (-3.70e4 + 6.53e3i)T + (5.58e8 - 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-2.42e4 + 1.40e4i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 3.53e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (3.87e4 - 4.62e4i)T + (-8.24e8 - 4.67e9i)T^{2} \) |
| 43 | \( 1 + (9.09e3 - 3.30e3i)T + (4.84e9 - 4.06e9i)T^{2} \) |
| 47 | \( 1 + (-1.60e3 - 9.13e3i)T + (-1.01e10 + 3.68e9i)T^{2} \) |
| 53 | \( 1 + (-7.61e4 + 2.09e5i)T + (-1.69e10 - 1.42e10i)T^{2} \) |
| 59 | \( 1 + (2.93e5 + 5.18e4i)T + (3.96e10 + 1.44e10i)T^{2} \) |
| 61 | \( 1 + (2.67e5 + 9.73e4i)T + (3.94e10 + 3.31e10i)T^{2} \) |
| 67 | \( 1 + (-2.13e5 + 3.76e4i)T + (8.50e10 - 3.09e10i)T^{2} \) |
| 71 | \( 1 + (8.26e4 + 2.27e5i)T + (-9.81e10 + 8.23e10i)T^{2} \) |
| 73 | \( 1 + (-5.18e5 - 4.35e5i)T + (2.62e10 + 1.49e11i)T^{2} \) |
| 79 | \( 1 + (4.79e5 - 5.71e5i)T + (-4.22e10 - 2.39e11i)T^{2} \) |
| 83 | \( 1 + (4.85e5 + 8.40e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-6.70e5 - 7.98e5i)T + (-8.62e10 + 4.89e11i)T^{2} \) |
| 97 | \( 1 + (-6.32e5 - 1.11e5i)T + (7.82e11 + 2.84e11i)T^{2} \) |
| 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
−17.19856186453208797800337052842, −15.35232305512159921274723833877, −14.20197734092567389159439878787, −13.22193300378096461470092860136, −12.50540503808504118083303283174, −10.09353503521478204154186220588, −8.425426143342380008914920432228, −6.23368792469685393080222741600, −4.84588391763411989196545532058, −2.21521995554484923079311325951,
2.80249605071985019616203754029, 4.59208263688949990150801859889, 6.34660488980057447418871832612, 8.967377412583541343341810595352, 10.30665712569535612886066415330, 12.10354972252805576042742153361, 13.76196233073765348445644652452, 14.14441317821829237043192687923, 15.46535583795657188083372966520, 17.29501133924807118597532450629
