L(s) = 1 | + (−0.866 − 0.5i)2-s + (1.15 − 1.29i)3-s + (0.499 + 0.866i)4-s + (2.17 − 0.520i)5-s + (−1.64 + 0.540i)6-s − 4.14i·7-s − 0.999i·8-s + (−0.333 − 2.98i)9-s + (−2.14 − 0.636i)10-s + 0.276i·11-s + (1.69 + 0.354i)12-s + (0.418 + 0.725i)13-s + (−2.07 + 3.58i)14-s + (1.83 − 3.40i)15-s + (−0.5 + 0.866i)16-s + (−2.82 + 4.90i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.666 − 0.745i)3-s + (0.249 + 0.433i)4-s + (0.972 − 0.232i)5-s + (−0.671 + 0.220i)6-s − 1.56i·7-s − 0.353i·8-s + (−0.111 − 0.993i)9-s + (−0.677 − 0.201i)10-s + 0.0834i·11-s + (0.489 + 0.102i)12-s + (0.116 + 0.201i)13-s + (−0.553 + 0.958i)14-s + (0.474 − 0.880i)15-s + (−0.125 + 0.216i)16-s + (−0.686 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.823068 - 1.28688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.823068 - 1.28688i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-1.15 + 1.29i)T \) |
| 5 | \( 1 + (-2.17 + 0.520i)T \) |
| 19 | \( 1 + (2.04 + 3.85i)T \) |
good | 7 | \( 1 + 4.14iT - 7T^{2} \) |
| 11 | \( 1 - 0.276iT - 11T^{2} \) |
| 13 | \( 1 + (-0.418 - 0.725i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.82 - 4.90i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.101 - 0.176i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.06 - 7.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.54iT - 31T^{2} \) |
| 37 | \( 1 - 1.84T + 37T^{2} \) |
| 41 | \( 1 + (-2.91 + 5.05i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.67 - 5.58i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (5.89 + 10.2i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.06 - 4.65i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.94 - 3.37i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.56 + 4.43i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.36 + 9.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.59 + 9.69i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.57 - 3.21i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-14.0 - 8.10i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.50T + 83T^{2} \) |
| 89 | \( 1 + (-3.27 - 5.67i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.24 + 5.62i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.55838467246098523228243157442, −9.442575988046434248090512543005, −8.799901025303188884236353254162, −7.898161533776881038956289988041, −6.87911458611010848877798919932, −6.39928581156767598093372008755, −4.60050110441494357825452376297, −3.40737660087330331865866188280, −2.06076095083955816509033802229, −1.03175590049426425026115341260,
2.15844320746517035512926350425, 2.81077895105604000379398364002, 4.62277071632049389950410998413, 5.68438819306209677830473334567, 6.30200188556263214734606838906, 7.77686357375553597352580842396, 8.570377035210503109295555152652, 9.426005185786779354358253474069, 9.670320151647384155598324027896, 10.77640575491276625345678176902