| 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
−15.11560971272791262356549760263, −13.93544726842356726697305316912, −13.49556327104145713224960116372, −11.23450130913131904353351546582, −10.03500346870309560047435925379, −9.403323425159832165077359977318, −7.12728543749050132637614537771, −6.63466729179953822338174401814, −3.30228991524847951394291938514, −2.31691843296835524965463819911,
2.24584608442742195105226760750, 3.85000898124855181127801963965, 6.19310066528004876338488115923, 8.244817797019475983178747165676, 8.748782729024116240688175153511, 9.763692700819240973497226503524, 12.29663829206464152698598427924, 12.86147527673340994559585277576, 13.92833702004731088608164625331, 15.52830887760110199662541456036
