L(s) = 1 | + (3.66 + 2.11i)2-s + (−23.0 − 39.9i)4-s + (64.9 − 37.5i)5-s + (181. − 313. i)7-s − 465. i·8-s + 317.·10-s + (1.48e3 + 856. i)11-s + (−1.35e3 − 2.35e3i)13-s + (1.32e3 − 765. i)14-s + (−492. + 852. i)16-s − 843. i·17-s + 3.68e3·19-s + (−2.99e3 − 1.73e3i)20-s + (3.61e3 + 6.26e3i)22-s + (−1.66e4 + 9.59e3i)23-s + ⋯ |
L(s) = 1 | + (0.457 + 0.264i)2-s + (−0.360 − 0.624i)4-s + (0.519 − 0.300i)5-s + (0.527 − 0.914i)7-s − 0.909i·8-s + 0.317·10-s + (1.11 + 0.643i)11-s + (−0.617 − 1.07i)13-s + (0.483 − 0.278i)14-s + (−0.120 + 0.208i)16-s − 0.171i·17-s + 0.536·19-s + (−0.374 − 0.216i)20-s + (0.339 + 0.588i)22-s + (−1.36 + 0.788i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(1.81126 - 0.885858i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81126 - 0.885858i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-3.66 - 2.11i)T + (32 + 55.4i)T^{2} \) |
| 5 | \( 1 + (-64.9 + 37.5i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-181. + 313. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.48e3 - 856. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.35e3 + 2.35e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 843. iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 3.68e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.66e4 - 9.59e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.69e4 - 1.55e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-4.35e3 - 7.53e3i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 3.21e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-5.17e4 + 2.98e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-6.08e4 + 1.05e5i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-7.99e4 - 4.61e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.34e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (5.21e4 - 3.01e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.13e5 - 1.96e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (3.48e4 + 6.03e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 1.85e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 8.10e4T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-2.46e4 + 4.26e4i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (6.13e5 + 3.54e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 3.64e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (8.21e4 - 1.42e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−15.62466624746175380222122839831, −14.36789242015742358510465672840, −13.70399346999656923319910745902, −12.27441946145724217683449096079, −10.42150882433929438203648077820, −9.400476223680039666973607763384, −7.35763123166585531873947604364, −5.65814070043247942877883210245, −4.26023277468938473104231859140, −1.16753896831638670830425704925,
2.36971387825857193495706041068, 4.34443480888145116160339273004, 6.15088400389975022498167200685, 8.239235605186544853623220268360, 9.483866376705133729117368354884, 11.53020605088262612490026087754, 12.22873663697929653419717995196, 13.92409288270230222787442613198, 14.46572941668267481129938744685, 16.32613367036917004594938853663