| L(s) = 1 | + (4 + 6.92i)2-s + (−24.5 − 42.4i)3-s + (−31.9 + 55.4i)4-s + (81.5 + 141. i)5-s + (196. − 339. i)6-s − 230.·7-s − 511.·8-s + (−109. + 189. i)9-s + (−652. + 1.12e3i)10-s + 7.72e3·11-s + 3.13e3·12-s + (6.13e3 − 1.06e4i)13-s + (−922. − 1.59e3i)14-s + (3.99e3 − 6.92e3i)15-s + (−2.04e3 − 3.54e3i)16-s + (9.68e3 + 1.67e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.524 − 0.908i)3-s + (−0.249 + 0.433i)4-s + (0.291 + 0.505i)5-s + (0.370 − 0.642i)6-s − 0.254·7-s − 0.353·8-s + (−0.0499 + 0.0864i)9-s + (−0.206 + 0.357i)10-s + 1.75·11-s + 0.524·12-s + (0.774 − 1.34i)13-s + (−0.0898 − 0.155i)14-s + (0.305 − 0.529i)15-s + (−0.125 − 0.216i)16-s + (0.478 + 0.828i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00917i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.98262 - 0.00909449i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98262 - 0.00909449i\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + (-2.92e4 + 6.04e3i)T \) |
| good | 3 | \( 1 + (24.5 + 42.4i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-81.5 - 141. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + 230.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 7.72e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-6.13e3 + 1.06e4i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-9.68e3 - 1.67e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 23 | \( 1 + (3.52e4 - 6.09e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-6.10e4 + 1.05e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 - 2.67e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.39e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (8.69e3 + 1.50e4i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.49e5 + 2.59e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (4.40e5 - 7.62e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (1.40e5 - 2.43e5i)T + (-5.87e11 - 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.60e5 + 2.77e5i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.38e6 + 2.39e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (7.36e5 - 1.27e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.23e6 - 2.13e6i)T + (-4.54e12 + 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.05e6 - 3.55e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (2.59e6 + 4.48e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 - 7.24e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-2.52e6 + 4.37e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.60e6 - 2.78e6i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.67104163579683470122017065439, −13.59817189813410209473547526145, −12.51496733029084075126463943655, −11.48346036233292463975032892472, −9.714571754075711349719868459609, −7.955154024581166886404658505168, −6.58161184197545910433370229212, −5.90235246946298349446841805741, −3.57327190302978587379892849356, −1.09414621904567048106726702246,
1.32399168913822680483637590012, 3.78054407288141869424675129659, 4.92318869741914469228910768559, 6.45971809447087384517453306890, 9.058914953551515953281419631928, 9.814622839239442574237178752082, 11.28277476922927273264788497469, 12.05371790752750518789528178717, 13.63383052416262376193893132118, 14.57958760305462575610443354989
