L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s − 5-s + (1.49 − 2.18i)7-s − 0.999i·8-s + (−0.866 + 0.5i)10-s − 3.47i·11-s + (0.972 − 0.561i)13-s + (0.205 − 2.63i)14-s + (−0.5 − 0.866i)16-s + (0.795 + 1.37i)17-s + (0.478 + 0.276i)19-s + (−0.499 + 0.866i)20-s + (−1.73 − 3.01i)22-s − 0.00487i·23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s − 0.447·5-s + (0.565 − 0.824i)7-s − 0.353i·8-s + (−0.273 + 0.158i)10-s − 1.04i·11-s + (0.269 − 0.155i)13-s + (0.0550 − 0.704i)14-s + (−0.125 − 0.216i)16-s + (0.192 + 0.334i)17-s + (0.109 + 0.0633i)19-s + (−0.111 + 0.193i)20-s + (−0.370 − 0.642i)22-s − 0.00101i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.551 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.132783594\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.132783594\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-1.49 + 2.18i)T \) |
good | 11 | \( 1 + 3.47iT - 11T^{2} \) |
| 13 | \( 1 + (-0.972 + 0.561i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.795 - 1.37i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.478 - 0.276i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 0.00487iT - 23T^{2} \) |
| 29 | \( 1 + (2.13 + 1.22i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.51 + 3.18i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.212 + 0.368i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.76 - 3.05i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.300 - 0.520i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.22 + 9.05i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.21 + 1.28i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.70 - 2.94i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.05 - 2.34i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.77 + 11.7i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.9iT - 71T^{2} \) |
| 73 | \( 1 + (4.45 - 2.57i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.37 - 4.11i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.21 - 5.57i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.59 + 6.23i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.3 - 8.26i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.904381155379190247074428831522, −8.060071351836250839067906066764, −7.44669667586469289271952308911, −6.44637000753088627662475957168, −5.63279964804009100382992106252, −4.75844883159420976258478076507, −3.82470452301239876058609017846, −3.29421136061088330119299067854, −1.86134102941120442484993750141, −0.63322791048566442654882726167,
1.67320345934151650582378081705, 2.73414622892777417213383293512, 3.81392612352224185467791501977, 4.72188241714245387273755386281, 5.33090210620634620078481089822, 6.24609781779946910690320946703, 7.21281956843566504011581778304, 7.71662922128813548693662993146, 8.659862810738610999638997765258, 9.276837623372591515478675492556