| L(s) = 1 | + (−3.80 − 1.23i)2-s + (−9.38 + 12.9i)3-s + (12.9 + 9.40i)4-s + (48.5 + 27.7i)5-s + (51.6 − 37.5i)6-s − 40.6i·7-s + (−37.6 − 51.7i)8-s + (−3.61 − 11.1i)9-s + (−150. − 165. i)10-s + (−48.2 + 148. i)11-s + (−242. + 78.9i)12-s + (−915. + 297. i)13-s + (−50.2 + 154. i)14-s + (−813. + 365. i)15-s + (79.1 + 243. i)16-s + (59.7 + 82.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 − 0.218i)2-s + (−0.601 + 0.828i)3-s + (0.404 + 0.293i)4-s + (0.867 + 0.497i)5-s + (0.585 − 0.425i)6-s − 0.313i·7-s + (−0.207 − 0.286i)8-s + (−0.0148 − 0.0458i)9-s + (−0.474 − 0.523i)10-s + (−0.120 + 0.370i)11-s + (−0.486 + 0.158i)12-s + (−1.50 + 0.488i)13-s + (−0.0685 + 0.211i)14-s + (−0.933 + 0.419i)15-s + (0.0772 + 0.237i)16-s + (0.0501 + 0.0690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.889 - 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.147477 + 0.608847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147477 + 0.608847i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.80 + 1.23i)T \) |
| 5 | \( 1 + (-48.5 - 27.7i)T \) |
| good | 3 | \( 1 + (9.38 - 12.9i)T + (-75.0 - 231. i)T^{2} \) |
| 7 | \( 1 + 40.6iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (48.2 - 148. i)T + (-1.30e5 - 9.46e4i)T^{2} \) |
| 13 | \( 1 + (915. - 297. i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-59.7 - 82.2i)T + (-4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (754. - 548. i)T + (7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 + (3.88e3 + 1.26e3i)T + (5.20e6 + 3.78e6i)T^{2} \) |
| 29 | \( 1 + (868. + 630. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.82e3 + 2.05e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-4.21e3 + 1.36e3i)T + (5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-3.49e3 - 1.07e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.06e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (761. - 1.04e3i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.03e4 - 2.80e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.28e4 + 3.96e4i)T + (-5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.18e4 - 3.65e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-1.03e3 - 1.42e3i)T + (-4.17e8 + 1.28e9i)T^{2} \) |
| 71 | \( 1 + (1.47e3 + 1.07e3i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-6.02e4 - 1.95e4i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-5.72e4 - 4.16e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (5.50e3 + 7.57e3i)T + (-1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-1.65e3 + 5.10e3i)T + (-4.51e9 - 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-4.84e4 + 6.66e4i)T + (-2.65e9 - 8.16e9i)T^{2} \) |
| 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
−15.16587958843354562723568823877, −14.04661970930790561958277237852, −12.46541572875991816787164402455, −11.14141690504892409578671934079, −10.09317786622058868306907857023, −9.642072344119560510339152242563, −7.66112877016200249521502163980, −6.16159346290190785544613238789, −4.51947876758600677956246046190, −2.23839726888191791521126605744,
0.40075868676011133515967451223, 2.08853252979901573278312973021, 5.35199481917987501124105233385, 6.42257174028475816146141057315, 7.76785471640101422726943568343, 9.223922185631718398649376826927, 10.29884577896284144620252048095, 11.90394762929334348607350609717, 12.68299847377170686542972581173, 13.98279101385142206402626939241
