L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−1.60 − 1.55i)5-s + (0.499 + 0.866i)6-s + 3.53i·7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.618 − 2.14i)10-s + 3.34·11-s + 0.999i·12-s + (1.48 − 0.854i)13-s + (−1.76 + 3.06i)14-s + (−0.618 − 2.14i)15-s + (−0.5 + 0.866i)16-s + (3.29 + 1.90i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.719 − 0.694i)5-s + (0.204 + 0.353i)6-s + 1.33i·7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.195 − 0.679i)10-s + 1.00·11-s + 0.288i·12-s + (0.410 − 0.237i)13-s + (−0.472 + 0.819i)14-s + (−0.159 − 0.554i)15-s + (−0.125 + 0.216i)16-s + (0.798 + 0.460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.76514 + 1.37203i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76514 + 1.37203i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (1.60 + 1.55i)T \) |
| 19 | \( 1 + (-1.30 - 4.15i)T \) |
good | 7 | \( 1 - 3.53iT - 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 + (-1.48 + 0.854i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.29 - 1.90i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (7.33 - 4.23i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.97 + 6.89i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.07T + 31T^{2} \) |
| 37 | \( 1 + 4.70iT - 37T^{2} \) |
| 41 | \( 1 + (-5.96 + 10.3i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.39 - 2.53i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.73 - 2.73i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.8 + 6.25i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.93 + 5.08i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.92 + 11.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.82 - 4.51i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.04 + 7.00i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.11 + 2.95i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.53 - 6.13i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.68iT - 83T^{2} \) |
| 89 | \( 1 + (-4.66 - 8.07i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.06 + 2.92i)T + (48.5 + 84.0i)T^{2} \) |
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
−11.23169784403058468626418453181, −9.772459540818616684672620251470, −9.038401723529222164786402871202, −8.177358354425803440647229868728, −7.62042133106987167078898349128, −5.99569424298633120452913948898, −5.52556246021839400468680015991, −4.09248992308455009985318158115, −3.55287046100137211936146618332, −1.92629532654439267617418932374,
1.12906673731350389457720736591, 2.85934769513652608281318332075, 3.82733431178586181918262707747, 4.43101357309760504236248953878, 6.16584292392040430363368641742, 7.04136318619879758847038281574, 7.57951902351156428921055847672, 8.780879648087726333103242145534, 9.930219447542802771321743483292, 10.64973950017767220725553525305