| L(s) = 1 | + (2.95 − 9.09i)2-s + (14.4 − 5.95i)3-s + (−48.0 − 34.8i)4-s + (−91.6 + 29.7i)5-s + (−11.5 − 148. i)6-s + (70.5 − 97.1i)7-s + (−211. + 153. i)8-s + (172. − 171. i)9-s + 921. i·10-s + (277. − 289. i)11-s + (−899. − 216. i)12-s + (390. + 126. i)13-s + (−674. − 928. i)14-s + (−1.14e3 + 974. i)15-s + (185. + 571. i)16-s + (39.0 + 120. i)17-s + ⋯ |
| L(s) = 1 | + (0.522 − 1.60i)2-s + (0.924 − 0.382i)3-s + (−1.50 − 1.09i)4-s + (−1.63 + 0.532i)5-s + (−0.131 − 1.68i)6-s + (0.544 − 0.749i)7-s + (−1.16 + 0.849i)8-s + (0.708 − 0.706i)9-s + 2.91i·10-s + (0.692 − 0.721i)11-s + (−1.80 − 0.434i)12-s + (0.640 + 0.208i)13-s + (−0.919 − 1.26i)14-s + (−1.31 + 1.11i)15-s + (0.181 + 0.557i)16-s + (0.0327 + 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.239339 - 2.04091i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239339 - 2.04091i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-14.4 + 5.95i)T \) |
| 11 | \( 1 + (-277. + 289. i)T \) |
| good | 2 | \( 1 + (-2.95 + 9.09i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (91.6 - 29.7i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-70.5 + 97.1i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-390. - 126. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-39.0 - 120. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.06e3 - 1.46e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 379. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-582. - 423. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (2.34e3 - 7.22e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (9.26e3 + 6.73e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-6.79e3 + 4.93e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 7.79e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.57e3 + 2.17e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.56e4 - 8.31e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (7.74e3 - 1.06e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (5.20e3 - 1.69e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 4.77e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (8.27e3 - 2.68e3i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (8.64e3 - 1.18e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (5.41e4 + 1.75e4i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (1.54e4 + 4.75e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 6.99e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (5.48e3 - 1.68e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.46502597519553287412354244971, −13.92309703626109616533230005367, −12.41592420371452816422279455318, −11.50982871607482558759681683990, −10.54721344555596402489955367699, −8.711584112307300286416611215184, −7.35629324940261493367357317198, −4.02318757426502425025734179829, −3.38423450733920810395483784993, −1.15064566431531651145941502413,
3.89636323270631711359115335638, 4.96531714467178209812138517318, 7.21428173380322641682920297367, 8.197875787771331652708292661243, 9.003909926054227278844932502929, 11.63646653204515946561537651947, 13.01003494133617150374653712622, 14.38830449856791756132555002214, 15.42835291931245645209116236742, 15.57136687976685401660325489956
