| L(s) = 1 | + (−10.5 + 4.10i)2-s + (1.43 − 3.47i)3-s + (94.2 − 86.6i)4-s + (−25.4 + 10.5i)5-s + (−0.891 + 42.5i)6-s + (−192. + 192. i)7-s + (−637. + 1.30e3i)8-s + (1.53e3 + 1.53e3i)9-s + (225. − 215. i)10-s + (−1.28e3 − 3.10e3i)11-s + (−165. − 451. i)12-s + (−1.09e4 − 4.54e3i)13-s + (1.24e3 − 2.82e3i)14-s + 103. i·15-s + (1.37e3 − 1.63e4i)16-s − 2.51e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.931 + 0.363i)2-s + (0.0307 − 0.0742i)3-s + (0.736 − 0.676i)4-s + (−0.0910 + 0.0377i)5-s + (−0.00168 + 0.0803i)6-s + (−0.212 + 0.212i)7-s + (−0.439 + 0.897i)8-s + (0.702 + 0.702i)9-s + (0.0711 − 0.0682i)10-s + (−0.291 − 0.703i)11-s + (−0.0276 − 0.0754i)12-s + (−1.38 − 0.573i)13-s + (0.120 − 0.275i)14-s + 0.00792i·15-s + (0.0837 − 0.996i)16-s − 1.24i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.137270 - 0.292748i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137270 - 0.292748i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.5 - 4.10i)T \) |
| good | 3 | \( 1 + (-1.43 + 3.47i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (25.4 - 10.5i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (192. - 192. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (1.28e3 + 3.10e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (1.09e4 + 4.54e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 2.51e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (2.01e4 + 8.35e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (4.83e4 + 4.83e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (-3.49e3 + 8.42e3i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 6.39e3T + 2.75e10T^{2} \) |
| 37 | \( 1 + (2.19e5 - 9.07e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (3.19e3 + 3.19e3i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (2.27e5 + 5.49e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 6.59e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-3.97e4 - 9.60e4i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-8.86e5 + 3.67e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (7.49e5 - 1.81e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-1.27e6 + 3.07e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-1.65e6 + 1.65e6i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-3.51e6 - 3.51e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 - 1.92e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-1.64e6 - 6.82e5i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (7.63e5 - 7.63e5i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 5.88e6T + 8.07e13T^{2} \) |
| 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
−15.19688672951355458876560176609, −13.76431893388704745395272891182, −12.15696200730591484594609323998, −10.69124972332899970784481490611, −9.641048681043959994742973429101, −8.124799928081300369605219408026, −7.00767309442577138806435158458, −5.24001369271026260742164248858, −2.38894500209783709989149787584, −0.19121839793817450760647717040,
1.90587322301080298348482199804, 4.00829145841903587554380794890, 6.64487652408198794846679174073, 7.895919463284572768935822779520, 9.544853509121333249657007793832, 10.28558315539777212807576378211, 11.98046715806027073776292461116, 12.78274168892139450549914126279, 14.79684853289522092268917034295, 15.85587091816772435446230791199
