| L(s) = 1 | + (8.24 + 1.45i)2-s + (−174. − 63.5i)4-s + (−723. − 861. i)5-s + (2.19e3 − 797. i)7-s + (−3.20e3 − 1.85e3i)8-s + (−4.71e3 − 8.15e3i)10-s + (−2.79e3 + 3.33e3i)11-s + (−6.27e3 − 3.55e4i)13-s + (1.92e4 − 3.39e3i)14-s + (1.27e4 + 1.06e4i)16-s + (−3.23e4 + 1.86e4i)17-s + (3.43e4 − 5.94e4i)19-s + (7.15e4 + 1.96e5i)20-s + (−2.79e4 + 2.34e4i)22-s + (−1.48e5 + 4.08e5i)23-s + ⋯ |
| L(s) = 1 | + (0.515 + 0.0908i)2-s + (−0.682 − 0.248i)4-s + (−1.15 − 1.37i)5-s + (0.912 − 0.332i)7-s + (−0.782 − 0.451i)8-s + (−0.471 − 0.815i)10-s + (−0.191 + 0.227i)11-s + (−0.219 − 1.24i)13-s + (0.500 − 0.0882i)14-s + (0.193 + 0.162i)16-s + (−0.387 + 0.223i)17-s + (0.263 − 0.456i)19-s + (0.446 + 1.22i)20-s + (−0.119 + 0.100i)22-s + (−0.531 + 1.46i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0804519 + 0.136894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0804519 + 0.136894i\) |
| \(L(5)\) |
|
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 | \( 1 + (-8.24 - 1.45i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (723. + 861. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (-2.19e3 + 797. i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (2.79e3 - 3.33e3i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (6.27e3 + 3.55e4i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (3.23e4 - 1.86e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-3.43e4 + 5.94e4i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.48e5 - 4.08e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (2.92e5 + 5.16e4i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (-7.64e5 - 2.78e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (1.13e6 + 1.96e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (1.45e6 - 2.56e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (-1.39e6 - 1.17e6i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (-6.47e5 - 1.78e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 - 6.52e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (1.53e7 + 1.82e7i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (-1.95e7 + 7.11e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (3.70e5 + 2.10e6i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (-1.54e7 + 8.91e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.66e7 - 2.88e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (9.65e6 - 5.47e7i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (-5.73e7 - 1.01e7i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (-4.54e7 - 2.62e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (7.13e7 + 5.98e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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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
−12.27177124999178576256756791291, −11.12287794395745847438136703533, −9.565782844933948139089364957127, −8.397337697793323305510371901259, −7.65887139547119826308574472019, −5.42329109937357534557753526523, −4.72078952646791806104811816837, −3.69384386087746535194721510364, −1.13431110774556468715051610941, −0.04804809051175890838460900581,
2.47917418039239714858229555993, 3.79408048631708634722758721720, 4.75358284589971051663627773362, 6.50126241434690073585006148464, 7.79536056628410219505950706001, 8.721540477438393782055058883481, 10.39185029998079996834005922659, 11.64750829625557098886968445892, 11.98215955052390347815586381822, 13.68743454986962400259801978088
