| L(s) = 1 | + (5.68 − 3.28i)2-s + (−3.48 + 2.01i)3-s + (13.5 − 23.4i)4-s + (5.86 + 10.1i)5-s + (−13.2 + 22.8i)6-s − 44.4·7-s − 72.7i·8-s + (−32.3 + 56.1i)9-s + (66.6 + 38.4i)10-s + 186.·11-s + 109. i·12-s + (−174. − 101. i)13-s + (−252. + 145. i)14-s + (−40.8 − 23.6i)15-s + (−22.0 − 38.1i)16-s + (−166. − 289. i)17-s + ⋯ |
| L(s) = 1 | + (1.42 − 0.820i)2-s + (−0.387 + 0.223i)3-s + (0.846 − 1.46i)4-s + (0.234 + 0.406i)5-s + (−0.367 + 0.635i)6-s − 0.906·7-s − 1.13i·8-s + (−0.399 + 0.692i)9-s + (0.666 + 0.384i)10-s + 1.54·11-s + 0.757i·12-s + (−1.03 − 0.597i)13-s + (−1.28 + 0.743i)14-s + (−0.181 − 0.104i)15-s + (−0.0860 − 0.148i)16-s + (−0.577 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.93386 - 0.770242i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.93386 - 0.770242i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-221. + 285. i)T \) |
| good | 2 | \( 1 + (-5.68 + 3.28i)T + (8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (3.48 - 2.01i)T + (40.5 - 70.1i)T^{2} \) |
| 5 | \( 1 + (-5.86 - 10.1i)T + (-312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + 44.4T + 2.40e3T^{2} \) |
| 11 | \( 1 - 186.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (174. + 101. i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (166. + 289. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 23 | \( 1 + (170. - 295. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-92.2 - 53.2i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 - 1.08e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 664. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (973. - 562. i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-1.67e3 - 2.90e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-407. + 706. i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + (2.48e3 + 1.43e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-1.46e3 + 847. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-910. + 1.57e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (5.41e3 + 3.12e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-1.85e3 + 1.07e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-3.21e3 - 5.56e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-6.64e3 + 3.83e3i)T + (1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 1.61e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (7.26e3 + 4.19e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-5.29e3 + 3.05e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
| 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
−17.52495127328500500226551146553, −16.02962969230314289680457178305, −14.51630042855497162106736175818, −13.60121954119832018915110420175, −12.20463317865043118810325111754, −11.17650320372238118677937134781, −9.746156066308192308934111435050, −6.53907450750488461148970737899, −4.90362465149445099359028329392, −2.92284834520781993355028211720,
3.93003981229310913139853285790, 5.89986264064007621598236466243, 6.86048522745591039438143555157, 9.319953329010519995603378229312, 11.94327468691302564288814755157, 12.65411679135048751230276476174, 14.08718917861576159703778826741, 15.05233971848498729927232276025, 16.60252502299223731388706201395, 17.17827241463425277718126761016
