| L(s) = 1 | + (6.42 + 2.33i)2-s + (21.8 − 41.3i)3-s + (−62.2 − 52.2i)4-s + (40.2 − 228. i)5-s + (237. − 214. i)6-s + (−1.34e3 + 1.12e3i)7-s + (−715. − 1.23e3i)8-s + (−1.23e3 − 1.80e3i)9-s + (793. − 1.37e3i)10-s + (151. + 860. i)11-s + (−3.51e3 + 1.43e3i)12-s + (8.11e3 − 2.95e3i)13-s + (−1.12e4 + 4.11e3i)14-s + (−8.56e3 − 6.65e3i)15-s + (105. + 599. i)16-s + (1.90e4 − 3.30e4i)17-s + ⋯ |
| L(s) = 1 | + (0.568 + 0.206i)2-s + (0.466 − 0.884i)3-s + (−0.486 − 0.407i)4-s + (0.144 − 0.817i)5-s + (0.448 − 0.405i)6-s + (−1.48 + 1.24i)7-s + (−0.494 − 0.855i)8-s + (−0.564 − 0.825i)9-s + (0.250 − 0.434i)10-s + (0.0343 + 0.194i)11-s + (−0.587 + 0.239i)12-s + (1.02 − 0.372i)13-s + (−1.10 + 0.400i)14-s + (−0.655 − 0.508i)15-s + (0.00644 + 0.0365i)16-s + (0.941 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.678001 - 1.45937i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.678001 - 1.45937i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-21.8 + 41.3i)T \) |
| good | 2 | \( 1 + (-6.42 - 2.33i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (-40.2 + 228. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (1.34e3 - 1.12e3i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (-151. - 860. i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (-8.11e3 + 2.95e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-1.90e4 + 3.30e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.45e4 + 2.51e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (228. + 191. i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-1.25e5 - 4.55e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (5.74e4 + 4.82e4i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (1.92e4 - 3.33e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.34e5 + 4.89e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (8.13e4 + 4.61e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (7.78e5 - 6.53e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 - 6.19e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (1.48e5 - 8.40e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (1.41e6 - 1.19e6i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-3.11e6 + 1.13e6i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-1.32e6 + 2.29e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (3.07e5 + 5.32e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-5.78e6 - 2.10e6i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (4.69e6 + 1.70e6i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (-1.66e6 - 2.88e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-5.56e4 - 3.15e5i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−15.22983427608317022406722819014, −13.75889157471032178719259893060, −12.93207383714350143959406039479, −12.20468968063273079983033802242, −9.493165982484581347324294522973, −8.789123303212157192713382766693, −6.56993398189679168142849927431, −5.41054372683560796906652518936, −3.06926052923468877734174686052, −0.65964812376697659128085462315,
3.30766717469421775072821620773, 3.93022141217320871591809992781, 6.25128947513338406887721632968, 8.276734777414860586770156572466, 9.895083150617454037233392968653, 10.78307195692820298489885962808, 12.79140261489071125935347345605, 13.81566557450336884064212194632, 14.63324290606493383410272857477, 16.23719110735651674094798158368
