| L(s) = 1 | + (1.02 + 0.969i)2-s + (1.72 − 0.103i)3-s + (0.118 + 1.99i)4-s + (−1.27 − 1.45i)5-s + (1.88 + 1.57i)6-s + (1.91 + 0.252i)7-s + (−1.81 + 2.16i)8-s + (2.97 − 0.358i)9-s + (0.0976 − 2.72i)10-s + (1.15 + 2.34i)11-s + (0.412 + 3.43i)12-s + (1.22 + 3.59i)13-s + (1.72 + 2.11i)14-s + (−2.35 − 2.37i)15-s + (−3.97 + 0.473i)16-s + (−3.52 − 3.52i)17-s + ⋯ |
| L(s) = 1 | + (0.727 + 0.685i)2-s + (0.998 − 0.0598i)3-s + (0.0593 + 0.998i)4-s + (−0.569 − 0.649i)5-s + (0.767 + 0.641i)6-s + (0.724 + 0.0953i)7-s + (−0.641 + 0.767i)8-s + (0.992 − 0.119i)9-s + (0.0308 − 0.863i)10-s + (0.349 + 0.707i)11-s + (0.118 + 0.992i)12-s + (0.338 + 0.997i)13-s + (0.461 + 0.566i)14-s + (−0.607 − 0.614i)15-s + (−0.992 + 0.118i)16-s + (−0.855 − 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.48211 + 1.47495i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48211 + 1.47495i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.02 - 0.969i)T \) |
| 3 | \( 1 + (-1.72 + 0.103i)T \) |
| good | 5 | \( 1 + (1.27 + 1.45i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-1.91 - 0.252i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (-1.15 - 2.34i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-1.22 - 3.59i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (3.52 + 3.52i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.42 + 7.16i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 9.09i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (3.39 - 0.222i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-5.33 + 3.08i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.105 - 0.530i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (0.336 + 2.55i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (8.62 - 4.25i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (0.702 - 0.188i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.11 - 3.17i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (9.51 + 10.8i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.426 + 6.50i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-4.48 - 2.21i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (4.61 + 11.1i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (0.143 - 0.345i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (0.696 + 2.59i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (11.4 + 10.0i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (3.73 - 1.54i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-13.7 - 7.93i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.46110741990388991587486676274, −9.412772966061471125384583481546, −9.033547298279061497345558807555, −8.065408780803647819480982655746, −7.34771392793387847505071840607, −6.59474617107108112981671082928, −4.79038436616610017333161231987, −4.57802185023586956596099182923, −3.33252701751021978188931471044, −1.93357739916148115389188143188,
1.50822956917976696537260740302, 2.89359006393864557369452247140, 3.67437834879454034644960666331, 4.53176825032367721012218442575, 5.90839516107361982340101168759, 6.93340106484154530355754966632, 8.168761021535472724023763065555, 8.634940519749451786197736291136, 10.07392806813809830894554161776, 10.61782767617616845564200013806
