| L(s) = 1 | + (1.81 + 2.49i)2-s + (3.16 + 1.02i)3-s + (−0.466 + 1.43i)4-s + (−10.8 − 2.62i)5-s + (3.17 + 9.76i)6-s + 5.10i·7-s + (19.0 − 6.18i)8-s + (−12.8 − 9.34i)9-s + (−13.1 − 31.8i)10-s + (−9.41 + 6.84i)11-s + (−2.95 + 4.06i)12-s + (27.9 − 38.4i)13-s + (−12.7 + 9.24i)14-s + (−31.7 − 19.5i)15-s + (59.7 + 43.3i)16-s + (−80.1 + 26.0i)17-s + ⋯ |
| L(s) = 1 | + (0.640 + 0.882i)2-s + (0.609 + 0.198i)3-s + (−0.0583 + 0.179i)4-s + (−0.971 − 0.235i)5-s + (0.215 + 0.664i)6-s + 0.275i·7-s + (0.841 − 0.273i)8-s + (−0.476 − 0.346i)9-s + (−0.415 − 1.00i)10-s + (−0.258 + 0.187i)11-s + (−0.0711 + 0.0979i)12-s + (0.595 − 0.819i)13-s + (−0.243 + 0.176i)14-s + (−0.546 − 0.335i)15-s + (0.932 + 0.677i)16-s + (−1.14 + 0.371i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.45040 + 0.707529i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45040 + 0.707529i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (10.8 + 2.62i)T \) |
| good | 2 | \( 1 + (-1.81 - 2.49i)T + (-2.47 + 7.60i)T^{2} \) |
| 3 | \( 1 + (-3.16 - 1.02i)T + (21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 - 5.10iT - 343T^{2} \) |
| 11 | \( 1 + (9.41 - 6.84i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-27.9 + 38.4i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (80.1 - 26.0i)T + (3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-32.9 - 101. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-26.0 - 35.7i)T + (-3.75e3 + 1.15e4i)T^{2} \) |
| 29 | \( 1 + (69.8 - 214. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (50.5 + 155. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-76.3 + 105. i)T + (-1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (110. + 80.6i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 414. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-510. - 165. i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-422. - 137. i)T + (1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (654. + 475. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (467. - 339. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-129. + 42.1i)T + (2.43e5 - 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-179. + 551. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (158. + 217. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (70.8 - 218. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (833. - 270. i)T + (4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + (1.24e3 - 905. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-3.24 - 1.05i)T + (7.38e5 + 5.36e5i)T^{2} \) |
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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
−16.81560020776193371320831471308, −15.49215210001495430860344366634, −15.15426988551899579338704661593, −13.85036335957888546333482066935, −12.49864382064714886562923903838, −10.79947042319736663466663007745, −8.780655967928470681966137946180, −7.52685096585631248907369217595, −5.69074498363590169069785005761, −3.83445768235635285886017135935,
2.78090252359852719958923826954, 4.37640831560534788142148813683, 7.28975799085845331385742976813, 8.679277727152547306100700730624, 10.96507976285903168013231332754, 11.62318975323176584907838231159, 13.22789415859452775149694635511, 13.97897170023821567702166593984, 15.49631209335835430620826485611, 16.79910515153735256906000895584
