| L(s) = 1 | + (−9.04 − 3.29i)2-s + (−20.3 + 42.1i)3-s + (−27.0 − 22.7i)4-s + (82.6 − 468. i)5-s + (322. − 313. i)6-s + (−558. + 468. i)7-s + (786. + 1.36e3i)8-s + (−1.36e3 − 1.71e3i)9-s + (−2.29e3 + 3.96e3i)10-s + (1.14e3 + 6.51e3i)11-s + (1.50e3 − 678. i)12-s + (7.64e3 − 2.78e3i)13-s + (6.59e3 − 2.39e3i)14-s + (1.80e4 + 1.30e4i)15-s + (−1.84e3 − 1.04e4i)16-s + (−9.78e3 + 1.69e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.799 − 0.290i)2-s + (−0.434 + 0.900i)3-s + (−0.211 − 0.177i)4-s + (0.295 − 1.67i)5-s + (0.609 − 0.593i)6-s + (−0.615 + 0.516i)7-s + (0.542 + 0.940i)8-s + (−0.621 − 0.783i)9-s + (−0.724 + 1.25i)10-s + (0.260 + 1.47i)11-s + (0.251 − 0.113i)12-s + (0.965 − 0.351i)13-s + (0.641 − 0.233i)14-s + (1.38 + 0.995i)15-s + (−0.112 − 0.637i)16-s + (−0.482 + 0.836i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.592075 + 0.314979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.592075 + 0.314979i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (20.3 - 42.1i)T \) |
| good | 2 | \( 1 + (9.04 + 3.29i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (-82.6 + 468. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (558. - 468. i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (-1.14e3 - 6.51e3i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (-7.64e3 + 2.78e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (9.78e3 - 1.69e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-8.43e3 - 1.46e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-4.98e4 - 4.18e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (-1.70e5 - 6.20e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (5.59e4 + 4.69e4i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (-5.19e4 + 8.99e4i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (4.27e5 - 1.55e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-4.08e4 - 2.31e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-2.85e3 + 2.39e3i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 - 1.85e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (3.90e5 - 2.21e6i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (6.59e5 - 5.53e5i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (1.12e6 - 4.10e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (1.06e6 - 1.84e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.06e6 + 1.84e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (4.45e6 + 1.62e6i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-1.09e6 - 3.99e5i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (-1.66e6 - 2.87e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-1.26e6 - 7.20e6i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−16.24107010276013241898447574033, −15.10142953960903895173405019370, −13.18338778943194893466180358710, −12.02052271509082865595047403444, −10.31615150577915662025178771860, −9.335103525741997894332570249483, −8.634442082887021336811935358591, −5.67791969626836333688143666108, −4.48010191901145727723683047461, −1.25240132402494937844663508394,
0.55378569246286598864164943210, 3.15791420617078448575741784081, 6.42604434595089614267467138305, 7.05303343315902468097443245151, 8.666611819193657266006217614601, 10.42288881613706345165907213200, 11.38530563418901815421911407934, 13.44506584864658420603462930668, 13.90850668601251515007743196880, 15.99938687377012574358067473650
