L(s) = 1 | + (−2.44 + 4.23i)2-s + (4.04 + 7.00i)4-s + (21.3 − 37.0i)5-s + (43.2 − 122. i)7-s − 196.·8-s + (104. + 181. i)10-s + (−355. − 615. i)11-s + 885.·13-s + (411. + 481. i)14-s + (349. − 605. i)16-s + (350. + 607. i)17-s + (627. − 1.08e3i)19-s + 345.·20-s + 3.47e3·22-s + (523. − 906. i)23-s + ⋯ |
L(s) = 1 | + (−0.432 + 0.748i)2-s + (0.126 + 0.219i)4-s + (0.382 − 0.662i)5-s + (0.333 − 0.942i)7-s − 1.08·8-s + (0.330 + 0.572i)10-s + (−0.885 − 1.53i)11-s + 1.45·13-s + (0.561 + 0.657i)14-s + (0.341 − 0.591i)16-s + (0.294 + 0.509i)17-s + (0.399 − 0.691i)19-s + 0.193·20-s + 1.53·22-s + (0.206 − 0.357i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.273i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.962 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.41762 - 0.197285i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41762 - 0.197285i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-43.2 + 122. i)T \) |
good | 2 | \( 1 + (2.44 - 4.23i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-21.3 + 37.0i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (355. + 615. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 885.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-350. - 607. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-627. + 1.08e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-523. + 906. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 6.15e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (1.14e3 + 1.98e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (202. - 349. i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.78e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.46e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.05e4 + 1.83e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (53.6 + 92.8i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.22e4 - 3.85e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.16e4 + 2.01e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.33e3 - 5.77e3i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 2.51e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (4.73e3 + 8.20e3i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-1.32e4 + 2.29e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 7.49e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.60e4 - 2.78e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.55e5T + 8.58e9T^{2} \) |
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
−13.74768656303858319934077023083, −13.16354572872573909828543416782, −11.53317165782000312662677720281, −10.45829355915656376566771111354, −8.703502840673316166858228875972, −8.189736183154214346261771431426, −6.68885898116134413100007129236, −5.42317196018258466553962216442, −3.40359457452569665556106217238, −0.815280316097173741732453913993,
1.66693094890332641188782487930, 2.89974664570084167583647620332, 5.31868938362709709086369339041, 6.62862699214713072443855765288, 8.372002678532643089343562625063, 9.732017246765153082062072759509, 10.48874096009288516450101096682, 11.61593991443827032408818663063, 12.57178759089621464448134635304, 14.09149093873541031598996588206