| L(s) = 1 | + (1.38 − 7.87i)2-s + (28.3 + 23.7i)3-s + (−60.1 − 21.8i)4-s + (−54.5 + 19.8i)5-s + (226. − 190. i)6-s + (−356. − 616. i)7-s + (−256 + 443. i)8-s + (−142. − 805. i)9-s + (80.5 + 457. i)10-s + (3.64e3 − 6.30e3i)11-s + (−1.18e3 − 2.05e3i)12-s + (3.07e3 − 2.57e3i)13-s + (−5.35e3 + 1.94e3i)14-s + (−2.01e3 − 734. i)15-s + (3.13e3 + 2.63e3i)16-s + (320. − 1.81e3i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.606 + 0.508i)3-s + (−0.469 − 0.171i)4-s + (−0.195 + 0.0709i)5-s + (0.428 − 0.359i)6-s + (−0.392 − 0.679i)7-s + (−0.176 + 0.306i)8-s + (−0.0649 − 0.368i)9-s + (0.0254 + 0.144i)10-s + (0.824 − 1.42i)11-s + (−0.197 − 0.342i)12-s + (0.388 − 0.325i)13-s + (−0.521 + 0.189i)14-s + (−0.154 − 0.0561i)15-s + (0.191 + 0.160i)16-s + (0.0158 − 0.0896i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.932439 - 1.48192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.932439 - 1.48192i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 + 7.87i)T \) |
| 19 | \( 1 + (1.72e4 + 2.44e4i)T \) |
| good | 3 | \( 1 + (-28.3 - 23.7i)T + (379. + 2.15e3i)T^{2} \) |
| 5 | \( 1 + (54.5 - 19.8i)T + (5.98e4 - 5.02e4i)T^{2} \) |
| 7 | \( 1 + (356. + 616. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.64e3 + 6.30e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-3.07e3 + 2.57e3i)T + (1.08e7 - 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-320. + 1.81e3i)T + (-3.85e8 - 1.40e8i)T^{2} \) |
| 23 | \( 1 + (4.33e4 + 1.57e4i)T + (2.60e9 + 2.18e9i)T^{2} \) |
| 29 | \( 1 + (1.43e4 + 8.14e4i)T + (-1.62e10 + 5.89e9i)T^{2} \) |
| 31 | \( 1 + (-1.37e5 - 2.38e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 1.69e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-3.54e5 - 2.97e5i)T + (3.38e10 + 1.91e11i)T^{2} \) |
| 43 | \( 1 + (2.51e5 - 9.15e4i)T + (2.08e11 - 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-2.63e4 - 1.49e5i)T + (-4.76e11 + 1.73e11i)T^{2} \) |
| 53 | \( 1 + (-4.80e5 - 1.74e5i)T + (8.99e11 + 7.55e11i)T^{2} \) |
| 59 | \( 1 + (1.41e5 - 8.03e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-4.58e5 - 1.66e5i)T + (2.40e12 + 2.02e12i)T^{2} \) |
| 67 | \( 1 + (6.09e5 + 3.45e6i)T + (-5.69e12 + 2.07e12i)T^{2} \) |
| 71 | \( 1 + (-4.87e5 + 1.77e5i)T + (6.96e12 - 5.84e12i)T^{2} \) |
| 73 | \( 1 + (3.21e6 + 2.69e6i)T + (1.91e12 + 1.08e13i)T^{2} \) |
| 79 | \( 1 + (-3.78e6 - 3.17e6i)T + (3.33e12 + 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-1.89e6 - 3.27e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (5.43e6 - 4.56e6i)T + (7.68e12 - 4.35e13i)T^{2} \) |
| 97 | \( 1 + (1.44e6 - 8.20e6i)T + (-7.59e13 - 2.76e13i)T^{2} \) |
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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
−14.18189779354355565856370024738, −13.38317813639731622012291094907, −11.82080990877574349420366596227, −10.69498374375641018753911728727, −9.444076442935254390547681306414, −8.375589128991786492454548069158, −6.29174084585605564186356745946, −4.08365448317235614612001358622, −3.13189418314000373226174007132, −0.71008454212367874933655575563,
2.05284995279004252865753350139, 4.17613015354462464046254376181, 6.07973550399284000003999155807, 7.43946429769442780659600412640, 8.576921502016752319311212854140, 9.822874785942656442713023269741, 11.93871749017345361490003245658, 12.92548706198872099058143511900, 14.14300248558590811325842219582, 15.04889131471007995680695003300
