| L(s) = 1 | + (5.58 − 1.49i)2-s + (11.2 + 3.00i)3-s + (15.0 − 8.69i)4-s + (−22.0 − 11.7i)5-s + 67.1·6-s + (−38.6 + 30.0i)7-s + (5.69 − 5.69i)8-s + (46.8 + 27.0i)9-s + (−140. − 32.8i)10-s + (−7.11 − 12.3i)11-s + (195. − 52.3i)12-s + (195. − 195. i)13-s + (−170. + 225. i)14-s + (−211. − 198. i)15-s + (−115. + 200. i)16-s + (−10.4 + 39.0i)17-s + ⋯ |
| L(s) = 1 | + (1.39 − 0.373i)2-s + (1.24 + 0.334i)3-s + (0.941 − 0.543i)4-s + (−0.881 − 0.471i)5-s + 1.86·6-s + (−0.789 + 0.613i)7-s + (0.0889 − 0.0889i)8-s + (0.578 + 0.333i)9-s + (−1.40 − 0.328i)10-s + (−0.0587 − 0.101i)11-s + (1.35 − 0.363i)12-s + (1.15 − 1.15i)13-s + (−0.871 + 1.15i)14-s + (−0.942 − 0.883i)15-s + (−0.452 + 0.784i)16-s + (−0.0362 + 0.135i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.966 + 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.15896 - 0.413834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.15896 - 0.413834i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (22.0 + 11.7i)T \) |
| 7 | \( 1 + (38.6 - 30.0i)T \) |
| good | 2 | \( 1 + (-5.58 + 1.49i)T + (13.8 - 8i)T^{2} \) |
| 3 | \( 1 + (-11.2 - 3.00i)T + (70.1 + 40.5i)T^{2} \) |
| 11 | \( 1 + (7.11 + 12.3i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-195. + 195. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (10.4 - 39.0i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-222. - 128. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (22.4 + 83.9i)T + (-2.42e5 + 1.39e5i)T^{2} \) |
| 29 | \( 1 - 1.11e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (890. + 1.54e3i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-805. + 215. i)T + (1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 - 2.73e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (209. - 209. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (523. - 140. i)T + (4.22e6 - 2.43e6i)T^{2} \) |
| 53 | \( 1 + (3.25e3 + 871. i)T + (6.83e6 + 3.94e6i)T^{2} \) |
| 59 | \( 1 + (1.42e3 - 825. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (438. - 759. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (365. - 1.36e3i)T + (-1.74e7 - 1.00e7i)T^{2} \) |
| 71 | \( 1 - 1.62e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-5.00e3 - 1.34e3i)T + (2.45e7 + 1.41e7i)T^{2} \) |
| 79 | \( 1 + (890. + 514. i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (6.26e3 - 6.26e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (5.77e3 + 3.33e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-1.22e4 - 1.22e4i)T + 8.85e7iT^{2} \) |
| 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
−15.38253934919864036162912379815, −14.52454986516404903105881285125, −13.21531053800469150254876922062, −12.58189834114018992237811193800, −11.15316626406158274433014427422, −9.204906370358011506989320642976, −8.085805515864171567416575698344, −5.72227070205953417086848631945, −3.87178128586912308136508855856, −3.01331032119473998723650530790,
3.12270282706995015625565921434, 4.09448217314466049512440200381, 6.51859279363843988660061030452, 7.58033144862979627463202676077, 9.231485202774487963198469458069, 11.30095927610490737789614358239, 12.73416332481205806112957445952, 13.77853012329140454314512441319, 14.28376576486579754724688922205, 15.53522792135882513163919091991
