| L(s) = 1 | + (−0.493 − 0.179i)2-s + (−38.9 + 25.9i)3-s + (−97.8 − 82.0i)4-s + (−47.2 + 268. i)5-s + (23.8 − 5.81i)6-s + (932. − 782. i)7-s + (67.1 + 116. i)8-s + (839. − 2.01e3i)9-s + (71.4 − 123. i)10-s + (163. + 927. i)11-s + (5.93e3 + 654. i)12-s + (7.90e3 − 2.87e3i)13-s + (−600. + 218. i)14-s + (−5.11e3 − 1.16e4i)15-s + (2.82e3 + 1.60e4i)16-s + (1.02e4 − 1.76e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0436 − 0.0158i)2-s + (−0.831 + 0.554i)3-s + (−0.764 − 0.641i)4-s + (−0.169 + 0.959i)5-s + (0.0450 − 0.0109i)6-s + (1.02 − 0.862i)7-s + (0.0463 + 0.0803i)8-s + (0.384 − 0.923i)9-s + (0.0226 − 0.0391i)10-s + (0.0370 + 0.210i)11-s + (0.991 + 0.109i)12-s + (0.998 − 0.363i)13-s + (−0.0585 + 0.0213i)14-s + (−0.391 − 0.891i)15-s + (0.172 + 0.978i)16-s + (0.503 − 0.872i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.00362 - 0.321715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00362 - 0.321715i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (38.9 - 25.9i)T \) |
| good | 2 | \( 1 + (0.493 + 0.179i)T + (98.0 + 82.2i)T^{2} \) |
| 5 | \( 1 + (47.2 - 268. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (-932. + 782. i)T + (1.43e5 - 8.11e5i)T^{2} \) |
| 11 | \( 1 + (-163. - 927. i)T + (-1.83e7 + 6.66e6i)T^{2} \) |
| 13 | \( 1 + (-7.90e3 + 2.87e3i)T + (4.80e7 - 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-1.02e4 + 1.76e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (2.37e4 + 4.12e4i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-6.25e4 - 5.24e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (1.47e5 + 5.35e4i)T + (1.32e10 + 1.10e10i)T^{2} \) |
| 31 | \( 1 + (7.97e4 + 6.69e4i)T + (4.77e9 + 2.70e10i)T^{2} \) |
| 37 | \( 1 + (-1.08e5 + 1.87e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-7.08e5 + 2.58e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (5.12e4 + 2.90e5i)T + (-2.55e11 + 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-7.08e5 + 5.94e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 - 1.61e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.24e5 + 7.03e5i)T + (-2.33e12 - 8.51e11i)T^{2} \) |
| 61 | \( 1 + (7.68e5 - 6.45e5i)T + (5.45e11 - 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-2.70e5 + 9.84e4i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-3.31e5 + 5.74e5i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-1.60e6 - 2.78e6i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (3.85e5 + 1.40e5i)T + (1.47e13 + 1.23e13i)T^{2} \) |
| 83 | \( 1 + (9.30e6 + 3.38e6i)T + (2.07e13 + 1.74e13i)T^{2} \) |
| 89 | \( 1 + (-5.57e6 - 9.65e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-2.70e5 - 1.53e6i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−15.42117702897692345147884989640, −14.58703739325796335556240706661, −13.32575260302725872531325524150, −11.09770078603135782588436192080, −10.81609324406686488532116407215, −9.267779076521582780702749123723, −7.18767266362786868837809168671, −5.43516359974955632521638680100, −4.05408350479242307874794686020, −0.75320277537776164634816310648,
1.30114967708446401594749777924, 4.43458454268322099625790621231, 5.75747600983261563472441963797, 8.021525928218694983847687967912, 8.794877546493638609149343223301, 11.03638014146580376147609638133, 12.35444148874495654584518821189, 12.90171537198494760014226514736, 14.51536827990672134773973794940, 16.41915305664648178019096531952
