| L(s) = 1 | + (−20.1 − 7.32i)2-s + (46.7 − 2.24i)3-s + (253. + 212. i)4-s + (26.6 − 151. i)5-s + (−956. − 297. i)6-s + (−1.12e3 + 942. i)7-s + (−2.17e3 − 3.76e3i)8-s + (2.17e3 − 209. i)9-s + (−1.64e3 + 2.84e3i)10-s + (679. + 3.85e3i)11-s + (1.23e4 + 9.36e3i)12-s + (−3.14e3 + 1.14e3i)13-s + (2.95e4 − 1.07e4i)14-s + (906. − 7.12e3i)15-s + (8.80e3 + 4.99e4i)16-s + (−6.06e3 + 1.05e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 − 0.647i)2-s + (0.998 − 0.0479i)3-s + (1.97 + 1.66i)4-s + (0.0953 − 0.540i)5-s + (−1.80 − 0.561i)6-s + (−1.23 + 1.03i)7-s + (−1.49 − 2.59i)8-s + (0.995 − 0.0957i)9-s + (−0.520 + 0.900i)10-s + (0.153 + 0.872i)11-s + (2.05 + 1.56i)12-s + (−0.397 + 0.144i)13-s + (2.87 − 1.04i)14-s + (0.0693 − 0.544i)15-s + (0.537 + 3.04i)16-s + (−0.299 + 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.678846 + 0.327983i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678846 + 0.327983i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-46.7 + 2.24i)T \) |
| good | 2 | \( 1 + (20.1 + 7.32i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (-26.6 + 151. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (1.12e3 - 942. i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (-679. - 3.85e3i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (3.14e3 - 1.14e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (6.06e3 - 1.05e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-1.60e4 - 2.78e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-2.99e4 - 2.51e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (1.44e5 + 5.27e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (-1.92e5 - 1.61e5i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (2.18e5 - 3.78e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-7.06e4 + 2.57e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (8.06e4 + 4.57e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (5.04e5 - 4.23e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + 4.24e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.81e5 + 1.02e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (-1.35e6 + 1.13e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (2.16e6 - 7.87e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (1.66e6 - 2.88e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (9.67e4 + 1.67e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-5.61e5 - 2.04e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-3.41e5 - 1.24e5i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (2.76e6 + 4.79e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-2.89e5 - 1.63e6i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−16.15340919450131125747205189539, −15.23077267724242286465196757545, −12.86339288995768004845193719660, −12.09920778341330235174225578652, −10.00682192398366186448953075788, −9.358045435537599602588713472546, −8.401389349872257512054142811768, −6.91911435481546917459172359414, −3.10422785946958254899873992082, −1.68953137451603625819554914614,
0.59584781970560932153362713537, 2.88281620806234787375866697473, 6.63963279202339468240014642715, 7.44712647450588914936962334067, 8.969088101766899649701648292694, 9.873151926619253402747539675099, 10.90342081044105641184606757403, 13.46300405609163181517331587994, 14.75214203951090197078844985385, 15.95189966066731873342512721103
