| L(s) = 1 | + (7.27 + 8.66i)2-s + (−64.9 − 48.3i)3-s + (−22.2 + 126. i)4-s + (−403. − 1.10e3i)5-s + (−53.2 − 914. i)6-s + (−619. − 3.51e3i)7-s + (−1.25e3 + 724. i)8-s + (1.88e3 + 6.28e3i)9-s + (6.67e3 − 1.15e4i)10-s + (−7.11e3 + 1.95e4i)11-s + (7.54e3 − 7.11e3i)12-s + (2.16e4 + 1.81e4i)13-s + (2.59e4 − 3.09e4i)14-s + (−2.74e4 + 9.15e4i)15-s + (−1.53e4 − 5.60e3i)16-s + (−6.01e4 − 3.47e4i)17-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.802 − 0.597i)3-s + (−0.0868 + 0.492i)4-s + (−0.645 − 1.77i)5-s + (−0.0411 − 0.705i)6-s + (−0.258 − 1.46i)7-s + (−0.306 + 0.176i)8-s + (0.286 + 0.958i)9-s + (0.667 − 1.15i)10-s + (−0.485 + 1.33i)11-s + (0.363 − 0.343i)12-s + (0.758 + 0.636i)13-s + (0.675 − 0.805i)14-s + (−0.541 + 1.80i)15-s + (−0.234 − 0.0855i)16-s + (−0.720 − 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0100124 + 0.0232097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0100124 + 0.0232097i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.27 - 8.66i)T \) |
| 3 | \( 1 + (64.9 + 48.3i)T \) |
| good | 5 | \( 1 + (403. + 1.10e3i)T + (-2.99e5 + 2.51e5i)T^{2} \) |
| 7 | \( 1 + (619. + 3.51e3i)T + (-5.41e6 + 1.97e6i)T^{2} \) |
| 11 | \( 1 + (7.11e3 - 1.95e4i)T + (-1.64e8 - 1.37e8i)T^{2} \) |
| 13 | \( 1 + (-2.16e4 - 1.81e4i)T + (1.41e8 + 8.03e8i)T^{2} \) |
| 17 | \( 1 + (6.01e4 + 3.47e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (7.33e4 + 1.27e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-3.07e5 - 5.42e4i)T + (7.35e10 + 2.67e10i)T^{2} \) |
| 29 | \( 1 + (-2.37e5 - 2.83e5i)T + (-8.68e10 + 4.92e11i)T^{2} \) |
| 31 | \( 1 + (8.74e4 - 4.95e5i)T + (-8.01e11 - 2.91e11i)T^{2} \) |
| 37 | \( 1 + (1.45e5 - 2.52e5i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + (2.34e5 - 2.78e5i)T + (-1.38e12 - 7.86e12i)T^{2} \) |
| 43 | \( 1 + (-8.71e5 - 3.17e5i)T + (8.95e12 + 7.51e12i)T^{2} \) |
| 47 | \( 1 + (3.13e6 - 5.52e5i)T + (2.23e13 - 8.14e12i)T^{2} \) |
| 53 | \( 1 - 1.14e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-4.51e6 - 1.24e7i)T + (-1.12e14 + 9.43e13i)T^{2} \) |
| 61 | \( 1 + (2.27e6 + 1.29e7i)T + (-1.80e14 + 6.55e13i)T^{2} \) |
| 67 | \( 1 + (2.05e7 + 1.72e7i)T + (7.05e13 + 3.99e14i)T^{2} \) |
| 71 | \( 1 + (1.61e7 + 9.30e6i)T + (3.22e14 + 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.09e7 + 1.88e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.58e7 + 1.32e7i)T + (2.63e14 - 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-8.05e6 - 9.59e6i)T + (-3.91e14 + 2.21e15i)T^{2} \) |
| 89 | \( 1 + (6.09e7 - 3.51e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + (-1.24e8 - 4.51e7i)T + (6.00e15 + 5.03e15i)T^{2} \) |
| 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
−13.08336855283686089514075160669, −12.12807791239798743859871743053, −10.87905791056400884861345450932, −9.042036964775354111100947377047, −7.62678221106786775192040328365, −6.78671387056670746715558530377, −4.87691505077955818057332095276, −4.38009965338847418580295982424, −1.25347321485589206091544561974, −0.009197256914729123458217072580,
2.75251222111938434454976196780, 3.68723444492281639498389011225, 5.70821418162235409549246706872, 6.40545472165208942360285788294, 8.529288952899115962807859998501, 10.26394548494059577608444547265, 11.05018929318321842079488666875, 11.66548464758184046853709474614, 12.99916048857715579200992818514, 14.65052764151048346521851410340
