L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 0.999·6-s + (2 + 1.73i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (2.5 + 4.33i)11-s + (−0.499 + 0.866i)12-s + 13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + (0.499 + 0.866i)18-s + (−3.5 + 6.06i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.408·6-s + (0.755 + 0.654i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.753 + 1.30i)11-s + (−0.144 + 0.249i)12-s + 0.277·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + (0.117 + 0.204i)18-s + (−0.802 + 1.39i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.788109472\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.788109472\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (-6.5 + 11.2i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3 + 5.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (2 + 3.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10T + 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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
−10.10707055583706186173978654607, −9.100267704336247307921981004885, −8.300346981177860970219670897280, −7.38899760279329238110308322989, −6.33266396547368135898643772412, −5.57892636235429746558871784286, −4.60159893461207423913874254458, −3.70338451388937210747419161297, −2.13770032939468137589282869593, −1.51872408554304091446836241848,
0.814867990252770269833364469561, 2.84294318080270307053696080587, 4.07461360718282739088559403278, 4.58609756827196261315479236059, 5.70791873969953932658680505403, 6.41979782070643375466106831190, 7.32161497568378477150502503111, 8.362018409722813014355108036590, 8.872646549810631368164433471788, 9.899765909449641144356693723034