L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s − 1.73i·5-s + (0.866 − 0.499i)6-s + (−4.09 + 2.36i)7-s + 0.999i·8-s + (−0.499 − 0.866i)9-s + (0.866 − 1.49i)10-s + (4.09 + 2.36i)11-s + 0.999·12-s + (−3.59 + 0.232i)13-s − 4.73·14-s + (−1.49 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−2.59 − 4.5i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s − 0.774i·5-s + (0.353 − 0.204i)6-s + (−1.54 + 0.894i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.273 − 0.474i)10-s + (1.23 + 0.713i)11-s + 0.288·12-s + (−0.997 + 0.0643i)13-s − 1.26·14-s + (−0.387 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.630 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24124 + 0.0479530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24124 + 0.0479530i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (3.59 - 0.232i)T \) |
good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 7 | \( 1 + (4.09 - 2.36i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.59 + 4.5i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 + 0.633i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.09 + 1.90i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.53iT - 31T^{2} \) |
| 37 | \( 1 + (-2.59 - 1.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.401 - 0.232i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.09 - 5.36i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 1.26iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (12 - 6.92i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.40 + 4.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.29 + 5.36i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.09 + 4.09i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 11.6iT - 83T^{2} \) |
| 89 | \( 1 + (-2.19 - 1.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.19 + 3i)T + (48.5 - 84.0i)T^{2} \) |
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
−14.43106666018703551964040024444, −13.24434780089022654718595896838, −12.42864789776079111620075164178, −11.89914697614154461698159110569, −9.552376313823090477480147145279, −8.950812568288712034051387321784, −7.17877352072917162392944041289, −6.26484740831525931551073563134, −4.67328986939593676657511962972, −2.78365226814312808697877704458,
3.10639155598610243462545224339, 4.05794074943596429287036818958, 6.15874339313983288500300342785, 7.10699435434673673449135277964, 9.184375554998852900409044871602, 10.17494998809962064716079754732, 11.00628850508995611589443769635, 12.40716194884352239996051409917, 13.50523769679742012855965949556, 14.35055208016376764816169793016