| L(s) = 1 | + (7.08 + 1.24i)2-s + (14.6 + 79.6i)3-s + (−191. − 69.8i)4-s + (349. + 416. i)5-s + (4.36 + 582. i)6-s + (22.4 − 8.16i)7-s + (−2.86e3 − 1.65e3i)8-s + (−6.13e3 + 2.33e3i)9-s + (1.95e3 + 3.38e3i)10-s + (−1.54e4 + 1.84e4i)11-s + (2.75e3 − 1.63e4i)12-s + (−8.79e3 − 4.98e4i)13-s + (169. − 29.8i)14-s + (−2.80e4 + 3.39e4i)15-s + (2.18e4 + 1.82e4i)16-s + (−3.75e4 + 2.16e4i)17-s + ⋯ |
| L(s) = 1 | + (0.442 + 0.0780i)2-s + (0.181 + 0.983i)3-s + (−0.749 − 0.272i)4-s + (0.559 + 0.666i)5-s + (0.00336 + 0.449i)6-s + (0.00933 − 0.00339i)7-s + (−0.700 − 0.404i)8-s + (−0.934 + 0.356i)9-s + (0.195 + 0.338i)10-s + (−1.05 + 1.25i)11-s + (0.132 − 0.786i)12-s + (−0.308 − 1.74i)13-s + (0.00440 − 0.000775i)14-s + (−0.554 + 0.670i)15-s + (0.332 + 0.279i)16-s + (−0.449 + 0.259i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.208i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.977 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.114812 + 1.08825i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.114812 + 1.08825i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-14.6 - 79.6i)T \) |
| good | 2 | \( 1 + (-7.08 - 1.24i)T + (240. + 87.5i)T^{2} \) |
| 5 | \( 1 + (-349. - 416. i)T + (-6.78e4 + 3.84e5i)T^{2} \) |
| 7 | \( 1 + (-22.4 + 8.16i)T + (4.41e6 - 3.70e6i)T^{2} \) |
| 11 | \( 1 + (1.54e4 - 1.84e4i)T + (-3.72e7 - 2.11e8i)T^{2} \) |
| 13 | \( 1 + (8.79e3 + 4.98e4i)T + (-7.66e8 + 2.78e8i)T^{2} \) |
| 17 | \( 1 + (3.75e4 - 2.16e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (7.51e4 - 1.30e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.08e5 - 2.99e5i)T + (-5.99e10 - 5.03e10i)T^{2} \) |
| 29 | \( 1 + (-9.45e5 - 1.66e5i)T + (4.70e11 + 1.71e11i)T^{2} \) |
| 31 | \( 1 + (-7.36e5 - 2.68e5i)T + (6.53e11 + 5.48e11i)T^{2} \) |
| 37 | \( 1 + (-1.21e6 - 2.10e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (-2.52e6 + 4.45e5i)T + (7.50e12 - 2.73e12i)T^{2} \) |
| 43 | \( 1 + (2.24e6 + 1.88e6i)T + (2.02e12 + 1.15e13i)T^{2} \) |
| 47 | \( 1 + (5.69e5 + 1.56e6i)T + (-1.82e13 + 1.53e13i)T^{2} \) |
| 53 | \( 1 - 3.88e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (-6.87e6 - 8.19e6i)T + (-2.54e13 + 1.44e14i)T^{2} \) |
| 61 | \( 1 + (1.49e7 - 5.43e6i)T + (1.46e14 - 1.23e14i)T^{2} \) |
| 67 | \( 1 + (-1.45e6 - 8.24e6i)T + (-3.81e14 + 1.38e14i)T^{2} \) |
| 71 | \( 1 + (9.42e6 - 5.43e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-1.18e7 + 2.05e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (7.07e6 - 4.01e7i)T + (-1.42e15 - 5.18e14i)T^{2} \) |
| 83 | \( 1 + (5.20e7 + 9.17e6i)T + (2.11e15 + 7.70e14i)T^{2} \) |
| 89 | \( 1 + (-3.60e7 - 2.07e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (7.44e7 + 6.24e7i)T + (1.36e15 + 7.71e15i)T^{2} \) |
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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
−15.51316127974532446708083721352, −14.96170102127791150946741731177, −13.80895776262535979933329509295, −12.58820476132584489830033653217, −10.26778460761354654888578806269, −10.04926591774070964336316981539, −8.168268958977323828188635191828, −5.82840809178227838849201642530, −4.63065091698468139263113761567, −2.87766591058602841224412822758,
0.41020630140866335930003978516, 2.52973639240019890921816048891, 4.74428852716822834260359102870, 6.31125395367102801671870405718, 8.293550695778315738744521594839, 9.178253059833773219048897699377, 11.46236655993960419392175687832, 12.79200692224100302420196467925, 13.52860406264107627817004952696, 14.30202103098367918138962589817
