| L(s) = 1 | + (−1.28 − 0.587i)2-s + (−0.859 + 2.92i)3-s + (1.30 + 1.51i)4-s + (2.44 + 4.36i)5-s + (2.82 − 3.26i)6-s + (1.96 − 13.6i)7-s + (−0.796 − 2.71i)8-s + (−0.264 − 0.169i)9-s + (−0.584 − 7.04i)10-s + (6.64 − 3.03i)11-s + (−5.55 + 2.53i)12-s + (21.6 − 3.11i)13-s + (−10.5 + 16.4i)14-s + (−14.8 + 3.41i)15-s + (−0.569 + 3.95i)16-s + (−17.9 + 20.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 − 0.293i)2-s + (−0.286 + 0.976i)3-s + (0.327 + 0.377i)4-s + (0.489 + 0.872i)5-s + (0.471 − 0.543i)6-s + (0.280 − 1.95i)7-s + (−0.0996 − 0.339i)8-s + (−0.0293 − 0.0188i)9-s + (−0.0584 − 0.704i)10-s + (0.604 − 0.276i)11-s + (−0.462 + 0.211i)12-s + (1.66 − 0.239i)13-s + (−0.754 + 1.17i)14-s + (−0.991 + 0.227i)15-s + (−0.0355 + 0.247i)16-s + (−1.05 + 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.22476 + 0.391532i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22476 + 0.391532i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 + 0.587i)T \) |
| 5 | \( 1 + (-2.44 - 4.36i)T \) |
| 23 | \( 1 + (-9.09 - 21.1i)T \) |
| good | 3 | \( 1 + (0.859 - 2.92i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (-1.96 + 13.6i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (-6.64 + 3.03i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (-21.6 + 3.11i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (17.9 - 20.7i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (-3.89 + 3.37i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (-18.1 + 20.9i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (-14.0 + 4.12i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-25.9 - 16.6i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (-17.5 + 11.2i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (5.99 + 1.76i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 + 47.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (7.47 - 51.9i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (-5.58 - 38.8i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (0.144 + 0.491i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (-22.6 + 49.6i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (16.7 - 36.6i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (49.7 - 43.1i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-112. + 16.1i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-62.5 - 40.2i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-9.74 + 33.1i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (50.5 - 32.4i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−11.30692999095072932441664965308, −10.88137651549948736304866986400, −10.37750731025121398347632373612, −9.513648000374535474755193579707, −8.222501877593008759585492351308, −7.05008975419297206107612020092, −6.12908409240025735351221580868, −4.22352149924367869735709281543, −3.57450924480028849541536450990, −1.32426106758098256698318246369,
1.15841224834003554659919139014, 2.29370516374901158078162679685, 4.87251720951404185072387015701, 6.07775053214194631278180837262, 6.58245149668224233620441737225, 8.171384710901303260380135066601, 8.920155891409367938792422106642, 9.425505247521386762696657457764, 11.18919993029664484078477461760, 11.94075240298497197432447611727
