| L(s) = 1 | + (−2.18 − 0.385i)2-s + (−26.6 − 4.57i)3-s + (−55.5 − 20.2i)4-s + (71.1 + 84.8i)5-s + (56.3 + 20.2i)6-s + (318. − 115. i)7-s + (236. + 136. i)8-s + (687. + 243. i)9-s + (−122. − 212. i)10-s + (27.8 − 33.2i)11-s + (1.38e3 + 791. i)12-s + (248. + 1.41e3i)13-s + (−740. + 130. i)14-s + (−1.50e3 − 2.58e3i)15-s + (2.43e3 + 2.04e3i)16-s + (7.29e3 − 4.21e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.272 − 0.0481i)2-s + (−0.985 − 0.169i)3-s + (−0.867 − 0.315i)4-s + (0.569 + 0.678i)5-s + (0.260 + 0.0937i)6-s + (0.929 − 0.338i)7-s + (0.461 + 0.266i)8-s + (0.942 + 0.334i)9-s + (−0.122 − 0.212i)10-s + (0.0209 − 0.0249i)11-s + (0.801 + 0.458i)12-s + (0.113 + 0.642i)13-s + (−0.269 + 0.0475i)14-s + (−0.446 − 0.765i)15-s + (0.593 + 0.498i)16-s + (1.48 − 0.857i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.997217 + 0.152383i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.997217 + 0.152383i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (26.6 + 4.57i)T \) |
| good | 2 | \( 1 + (2.18 + 0.385i)T + (60.1 + 21.8i)T^{2} \) |
| 5 | \( 1 + (-71.1 - 84.8i)T + (-2.71e3 + 1.53e4i)T^{2} \) |
| 7 | \( 1 + (-318. + 115. i)T + (9.01e4 - 7.56e4i)T^{2} \) |
| 11 | \( 1 + (-27.8 + 33.2i)T + (-3.07e5 - 1.74e6i)T^{2} \) |
| 13 | \( 1 + (-248. - 1.41e3i)T + (-4.53e6 + 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-7.29e3 + 4.21e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (4.97e3 - 8.61e3i)T + (-2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (926. - 2.54e3i)T + (-1.13e8 - 9.51e7i)T^{2} \) |
| 29 | \( 1 + (-1.79e4 - 3.15e3i)T + (5.58e8 + 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-2.26e4 - 8.25e3i)T + (6.79e8 + 5.70e8i)T^{2} \) |
| 37 | \( 1 + (-3.68e4 - 6.38e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-9.35e4 + 1.64e4i)T + (4.46e9 - 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-2.90e4 - 2.43e4i)T + (1.09e9 + 6.22e9i)T^{2} \) |
| 47 | \( 1 + (2.58e4 + 7.09e4i)T + (-8.25e9 + 6.92e9i)T^{2} \) |
| 53 | \( 1 - 1.48e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (1.40e5 + 1.67e5i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (1.53e5 - 5.59e4i)T + (3.94e10 - 3.31e10i)T^{2} \) |
| 67 | \( 1 + (5.08e4 + 2.88e5i)T + (-8.50e10 + 3.09e10i)T^{2} \) |
| 71 | \( 1 + (1.92e5 - 1.10e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (2.65e5 - 4.59e5i)T + (-7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.40e5 + 7.99e5i)T + (-2.28e11 - 8.31e10i)T^{2} \) |
| 83 | \( 1 + (-7.63e5 - 1.34e5i)T + (3.07e11 + 1.11e11i)T^{2} \) |
| 89 | \( 1 + (-7.18e5 - 4.14e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (9.61e5 + 8.06e5i)T + (1.44e11 + 8.20e11i)T^{2} \) |
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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
−16.52687988965912555251102049769, −14.52446520150436548314431947522, −13.80251058993148112719645261645, −12.11415528366620084877644109438, −10.71197582184842033733813649397, −9.836844176553701233566630279705, −7.85658439098123406226973632912, −6.07172811213365668460024207439, −4.62351490545443985632426060662, −1.26688821346914554468830122702,
0.952742300094690322907616392485, 4.55037012826992820228198260852, 5.67666582239223446562926671056, 7.980777597395546190492472529864, 9.344645485716359620111526538484, 10.67418300822291866999290824076, 12.27630790860751519063541940200, 13.17918644838475769972798587930, 14.78476806212824057475414827347, 16.39177825517883018861062105538
