| L(s) = 1 | + (−0.317 + 0.659i)2-s + (14.0 + 1.05i)3-s + (9.64 + 12.0i)4-s + (13.2 − 42.9i)5-s + (−5.16 + 8.93i)6-s + (−57.8 + 33.3i)7-s + (−22.4 + 5.12i)8-s + (116. + 17.5i)9-s + (24.0 + 22.3i)10-s + (37.7 − 47.3i)11-s + (122. + 180. i)12-s + (14.0 − 13.0i)13-s + (−3.65 − 48.7i)14-s + (231. − 589. i)15-s + (−51.3 + 224. i)16-s + (−299. + 92.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.0793 + 0.164i)2-s + (1.56 + 0.117i)3-s + (0.602 + 0.755i)4-s + (0.529 − 1.71i)5-s + (−0.143 + 0.248i)6-s + (−1.17 + 0.681i)7-s + (−0.350 + 0.0800i)8-s + (1.44 + 0.217i)9-s + (0.240 + 0.223i)10-s + (0.312 − 0.391i)11-s + (0.853 + 1.25i)12-s + (0.0829 − 0.0769i)13-s + (−0.0186 − 0.248i)14-s + (1.02 − 2.62i)15-s + (−0.200 + 0.878i)16-s + (−1.03 + 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.37784 + 0.238972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37784 + 0.238972i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (832. + 1.65e3i)T \) |
| good | 2 | \( 1 + (0.317 - 0.659i)T + (-9.97 - 12.5i)T^{2} \) |
| 3 | \( 1 + (-14.0 - 1.05i)T + (80.0 + 12.0i)T^{2} \) |
| 5 | \( 1 + (-13.2 + 42.9i)T + (-516. - 352. i)T^{2} \) |
| 7 | \( 1 + (57.8 - 33.3i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-37.7 + 47.3i)T + (-3.25e3 - 1.42e4i)T^{2} \) |
| 13 | \( 1 + (-14.0 + 13.0i)T + (2.13e3 - 2.84e4i)T^{2} \) |
| 17 | \( 1 + (299. - 92.3i)T + (6.90e4 - 4.70e4i)T^{2} \) |
| 19 | \( 1 + (-88.7 - 588. i)T + (-1.24e5 + 3.84e4i)T^{2} \) |
| 23 | \( 1 + (258. + 658. i)T + (-2.05e5 + 1.90e5i)T^{2} \) |
| 29 | \( 1 + (-177. + 13.3i)T + (6.99e5 - 1.05e5i)T^{2} \) |
| 31 | \( 1 + (-617. + 421. i)T + (3.37e5 - 8.59e5i)T^{2} \) |
| 37 | \( 1 + (149. + 86.1i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (583. + 281. i)T + (1.76e6 + 2.20e6i)T^{2} \) |
| 47 | \( 1 + (-1.91e3 - 2.40e3i)T + (-1.08e6 + 4.75e6i)T^{2} \) |
| 53 | \( 1 + (523. + 485. i)T + (5.89e5 + 7.86e6i)T^{2} \) |
| 59 | \( 1 + (424. - 1.85e3i)T + (-1.09e7 - 5.25e6i)T^{2} \) |
| 61 | \( 1 + (-2.18e3 + 3.19e3i)T + (-5.05e6 - 1.28e7i)T^{2} \) |
| 67 | \( 1 + (-4.74e3 + 714. i)T + (1.92e7 - 5.93e6i)T^{2} \) |
| 71 | \( 1 + (-6.57e3 - 2.57e3i)T + (1.86e7 + 1.72e7i)T^{2} \) |
| 73 | \( 1 + (-2.72e3 - 2.93e3i)T + (-2.12e6 + 2.83e7i)T^{2} \) |
| 79 | \( 1 + (-180. - 312. i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-165. + 2.21e3i)T + (-4.69e7 - 7.07e6i)T^{2} \) |
| 89 | \( 1 + (2.33e3 + 174. i)T + (6.20e7 + 9.35e6i)T^{2} \) |
| 97 | \( 1 + (2.43e3 - 3.04e3i)T + (-1.96e7 - 8.63e7i)T^{2} \) |
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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
−15.52830887760110199662541456036, −13.92833702004731088608164625331, −12.86147527673340994559585277576, −12.29663829206464152698598427924, −9.763692700819240973497226503524, −8.748782729024116240688175153511, −8.244817797019475983178747165676, −6.19310066528004876338488115923, −3.85000898124855181127801963965, −2.24584608442742195105226760750,
2.31691843296835524965463819911, 3.30228991524847951394291938514, 6.63466729179953822338174401814, 7.12728543749050132637614537771, 9.403323425159832165077359977318, 10.03500346870309560047435925379, 11.23450130913131904353351546582, 13.49556327104145713224960116372, 13.93544726842356726697305316912, 15.11560971272791262356549760263
