| L(s) = 1 | + (1.23 + 3.80i)2-s + (2.72 − 8.39i)3-s + (−12.9 + 9.40i)4-s − 56.0·5-s + 35.3·6-s + (116. − 84.7i)7-s + (−51.7 − 37.6i)8-s + (133. + 97.0i)9-s + (−69.3 − 213. i)10-s + (371. − 269. i)11-s + (43.6 + 134. i)12-s + (304. − 936. i)13-s + (466. + 338. i)14-s + (−153. + 470. i)15-s + (79.1 − 243. i)16-s + (−624. − 453. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.175 − 0.538i)3-s + (−0.404 + 0.293i)4-s − 1.00·5-s + 0.400·6-s + (0.899 − 0.653i)7-s + (−0.286 − 0.207i)8-s + (0.549 + 0.399i)9-s + (−0.219 − 0.674i)10-s + (0.924 − 0.671i)11-s + (0.0875 + 0.269i)12-s + (0.499 − 1.53i)13-s + (0.636 + 0.462i)14-s + (−0.175 + 0.540i)15-s + (0.0772 − 0.237i)16-s + (−0.523 − 0.380i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.77546 - 0.446028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77546 - 0.446028i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 - 3.80i)T \) |
| 31 | \( 1 + (-925. - 5.27e3i)T \) |
| good | 3 | \( 1 + (-2.72 + 8.39i)T + (-196. - 142. i)T^{2} \) |
| 5 | \( 1 + 56.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + (-116. + 84.7i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 11 | \( 1 + (-371. + 269. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-304. + 936. i)T + (-3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (624. + 453. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (326. + 1.00e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-3.44e3 - 2.50e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (922. + 2.83e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 37 | \( 1 + 1.18e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-3.83e3 - 1.17e4i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + (5.76e3 + 1.77e4i)T + (-1.18e8 + 8.64e7i)T^{2} \) |
| 47 | \( 1 + (-3.62e3 + 1.11e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (9.37e3 + 6.80e3i)T + (1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (9.86e3 - 3.03e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 - 5.02e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.47e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (2.08e4 + 1.51e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-4.17e4 + 3.03e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.66e4 + 1.21e4i)T + (9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-2.73e4 - 8.43e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-5.52e4 + 4.01e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (6.36e4 - 4.62e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
| 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
−13.84540053822774472302990128825, −13.13136305665104261934160574822, −11.72703398064859661835185697333, −10.71000186128761925067787794359, −8.668665290025067655726651032758, −7.76848600628723907046503950100, −6.91069296893115003099844686399, −5.02339683670804890848209878591, −3.62059755226755654384078120756, −0.911129139889552437260941827293,
1.67881603862618039257311874192, 3.87585537360234659296090704514, 4.60427548492931448190968606089, 6.77368551488198155472965347872, 8.551680061392751275726598403165, 9.399527059632658790162509909852, 10.96609682165224737838343502403, 11.76527766354279216631560658423, 12.62734179236004856026881783692, 14.39413173345556817290743833688
