| L(s) = 1 | + (−0.0604 − 0.0572i)2-s + (1.46 − 0.930i)3-s + (−0.107 − 1.99i)4-s + (−3.23 + 0.529i)5-s + (−0.141 − 0.0273i)6-s + (−1.62 − 3.06i)7-s + (−0.215 + 0.253i)8-s + (1.26 − 2.71i)9-s + (0.225 + 0.152i)10-s + (3.14 + 0.342i)11-s + (−2.01 − 2.80i)12-s + (1.78 + 3.86i)13-s + (−0.0772 + 0.278i)14-s + (−4.22 + 3.78i)15-s + (−3.93 + 0.427i)16-s + (1.00 + 0.534i)17-s + ⋯ |
| L(s) = 1 | + (−0.0427 − 0.0404i)2-s + (0.843 − 0.537i)3-s + (−0.0539 − 0.995i)4-s + (−1.44 + 0.237i)5-s + (−0.0577 − 0.0111i)6-s + (−0.614 − 1.15i)7-s + (−0.0760 + 0.0895i)8-s + (0.422 − 0.906i)9-s + (0.0713 + 0.0483i)10-s + (0.949 + 0.103i)11-s + (−0.580 − 0.810i)12-s + (0.495 + 1.07i)13-s + (−0.0206 + 0.0743i)14-s + (−1.09 + 0.976i)15-s + (−0.983 + 0.106i)16-s + (0.244 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0803 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0803 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.775215 - 0.840231i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.775215 - 0.840231i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.46 + 0.930i)T \) |
| 59 | \( 1 + (-4.21 - 6.42i)T \) |
| good | 2 | \( 1 + (0.0604 + 0.0572i)T + (0.108 + 1.99i)T^{2} \) |
| 5 | \( 1 + (3.23 - 0.529i)T + (4.73 - 1.59i)T^{2} \) |
| 7 | \( 1 + (1.62 + 3.06i)T + (-3.92 + 5.79i)T^{2} \) |
| 11 | \( 1 + (-3.14 - 0.342i)T + (10.7 + 2.36i)T^{2} \) |
| 13 | \( 1 + (-1.78 - 3.86i)T + (-8.41 + 9.90i)T^{2} \) |
| 17 | \( 1 + (-1.00 - 0.534i)T + (9.54 + 14.0i)T^{2} \) |
| 19 | \( 1 + (-5.72 - 3.44i)T + (8.89 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-3.32 + 8.34i)T + (-16.6 - 15.8i)T^{2} \) |
| 29 | \( 1 + (-3.45 - 3.64i)T + (-1.57 + 28.9i)T^{2} \) |
| 31 | \( 1 + (1.85 + 3.09i)T + (-14.5 + 27.3i)T^{2} \) |
| 37 | \( 1 + (3.97 - 3.37i)T + (5.98 - 36.5i)T^{2} \) |
| 41 | \( 1 + (7.19 - 2.86i)T + (29.7 - 28.1i)T^{2} \) |
| 43 | \( 1 + (0.350 + 3.22i)T + (-41.9 + 9.24i)T^{2} \) |
| 47 | \( 1 + (0.457 - 2.79i)T + (-44.5 - 15.0i)T^{2} \) |
| 53 | \( 1 + (3.98 - 2.69i)T + (19.6 - 49.2i)T^{2} \) |
| 61 | \( 1 + (1.06 - 1.12i)T + (-3.30 - 60.9i)T^{2} \) |
| 67 | \( 1 + (3.77 + 3.20i)T + (10.8 + 66.1i)T^{2} \) |
| 71 | \( 1 + (-6.29 - 1.03i)T + (67.2 + 22.6i)T^{2} \) |
| 73 | \( 1 + (-15.0 - 4.16i)T + (62.5 + 37.6i)T^{2} \) |
| 79 | \( 1 + (4.05 - 0.892i)T + (71.6 - 33.1i)T^{2} \) |
| 83 | \( 1 + (5.19 - 3.95i)T + (22.2 - 79.9i)T^{2} \) |
| 89 | \( 1 + (3.52 - 3.33i)T + (4.81 - 88.8i)T^{2} \) |
| 97 | \( 1 + (-4.94 + 1.37i)T + (83.1 - 50.0i)T^{2} \) |
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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
−12.36805403548857681866293481011, −11.50486508758150238602293828835, −10.39989287249017994315962791026, −9.379942599424661663811826614216, −8.335577433603649421206326428822, −7.04419075890136514461399493202, −6.61937060074990568499784561815, −4.32863056428157758639587659297, −3.45265400066772091264884523006, −1.12462133436482515561373217613,
3.18342576320835198336145557103, 3.57887558893511847821148727141, 5.13813095309240376647989074176, 7.13300789227178823784894136575, 8.037549115397743718567615462164, 8.807212632239095601841487759853, 9.532415109978762032626828874303, 11.30748454871335477726748920107, 11.95603129311504755141098746525, 12.84719717202830249531633771790
