| L(s) = 1 | + (−8 + 13.8i)2-s + (−22.2 + 38.6i)3-s + (−127. − 221. i)4-s + (397. − 688. i)5-s + (−356. − 617. i)6-s + 2.27e3·7-s + 4.09e3·8-s + (8.84e3 + 1.53e4i)9-s + (6.35e3 + 1.10e4i)10-s − 8.41e4·11-s + 1.14e4·12-s + (4.64e4 + 8.03e4i)13-s + (−1.81e4 + 3.14e4i)14-s + (1.77e4 + 3.06e4i)15-s + (−3.27e4 + 5.67e4i)16-s + (1.09e5 − 1.89e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.158 + 0.275i)3-s + (−0.249 − 0.433i)4-s + (0.284 − 0.492i)5-s + (−0.112 − 0.194i)6-s + 0.357·7-s + 0.353·8-s + (0.449 + 0.778i)9-s + (0.201 + 0.348i)10-s − 1.73·11-s + 0.158·12-s + (0.450 + 0.780i)13-s + (−0.126 + 0.218i)14-s + (0.0903 + 0.156i)15-s + (−0.125 + 0.216i)16-s + (0.318 − 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0631882 - 0.255463i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0631882 - 0.255463i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8 - 13.8i)T \) |
| 19 | \( 1 + (5.47e5 - 1.52e5i)T \) |
| good | 3 | \( 1 + (22.2 - 38.6i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-397. + 688. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 - 2.27e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.41e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-4.64e4 - 8.03e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-1.09e5 + 1.89e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 23 | \( 1 + (6.64e5 + 1.15e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (3.09e6 + 5.36e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + 6.00e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.63e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (1.66e7 - 2.87e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (-1.35e7 + 2.35e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-1.42e7 - 2.46e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.57e7 - 4.46e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-1.03e7 + 1.79e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-6.82e7 - 1.18e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.50e8 + 2.61e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-3.31e6 + 5.73e6i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (7.50e7 - 1.29e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.70e8 - 2.96e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 2.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (2.15e8 + 3.73e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-1.72e8 + 2.99e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
| 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
−15.26193210361499802594767770168, −13.81231026558834236818747780361, −12.86859256042016556756875452193, −11.00841797038087696842860106762, −10.01067938683373067027011538054, −8.561798129009431292973879827246, −7.45456170506223074590366783188, −5.64029527633824933419101136930, −4.57726968727221802042017704939, −1.93132566014396175580832188209,
0.10412710238458323327605414553, 1.85459312996521613608203062162, 3.44559627431943740554133368859, 5.50138783827170259009984550715, 7.23359385955452259709252748126, 8.543138110904835826011469517085, 10.20731997052697827459969181424, 10.88592367838533012869414818484, 12.48061588488430380126249714245, 13.22842323113106782087610669701
