| L(s) = 1 | + (−4 + 6.92i)2-s + (−45.4 + 78.6i)3-s + (−31.9 − 55.4i)4-s + (149. − 259. i)5-s + (−363. − 629. i)6-s − 708.·7-s + 511.·8-s + (−3.03e3 − 5.25e3i)9-s + (1.19e3 + 2.07e3i)10-s + 952.·11-s + 5.81e3·12-s + (4.02e3 + 6.97e3i)13-s + (2.83e3 − 4.90e3i)14-s + (1.36e4 + 2.35e4i)15-s + (−2.04e3 + 3.54e3i)16-s + (9.35e3 − 1.62e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.971 + 1.68i)3-s + (−0.249 − 0.433i)4-s + (0.535 − 0.928i)5-s + (−0.687 − 1.18i)6-s − 0.780·7-s + 0.353·8-s + (−1.38 − 2.40i)9-s + (0.378 + 0.656i)10-s + 0.215·11-s + 0.971·12-s + (0.508 + 0.880i)13-s + (0.275 − 0.477i)14-s + (1.04 + 1.80i)15-s + (−0.125 + 0.216i)16-s + (0.461 − 0.800i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.612342 - 0.0418252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.612342 - 0.0418252i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 19 | \( 1 + (2.80e4 + 1.02e4i)T \) |
| good | 3 | \( 1 + (45.4 - 78.6i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-149. + 259. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 + 708.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 952.T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.02e3 - 6.97e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-9.35e3 + 1.62e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-1.20e4 - 2.08e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.05e5 + 1.83e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.72e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.73e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-4.04e5 + 7.00e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (-3.84e5 + 6.66e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-8.82e4 - 1.52e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-2.52e5 - 4.37e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.77e5 - 3.07e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.05e6 + 1.82e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (3.25e5 + 5.63e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-7.56e5 + 1.30e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-3.04e6 + 5.27e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.25e6 + 2.17e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 7.94e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.58e6 - 7.93e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (2.20e6 - 3.81e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−15.35447897824143080978368823756, −13.80921300031491231009659133739, −12.11447172354684761425108736149, −10.74262432952878298035416898417, −9.480387326215172165703944241604, −9.066850423935615037068480352927, −6.37884466130461876642588431568, −5.32908191323379501553900604102, −4.08543523403469281443084391887, −0.38394128180282753280913173423,
1.26011374193958150886702058712, 2.78123844054864875865859779031, 5.93192986200137552464760184374, 6.78290825076589555155993965123, 8.197802190141215844226637313966, 10.32154343278683787938422712898, 11.12765979027717880375050732116, 12.57210549952785277182010849110, 13.04651609851581827754031269867, 14.37680732594492814928184736437
