| L(s) = 1 | + (−12.1 + 8.82i)2-s + (−8.34 − 25.6i)3-s + (30.0 − 92.4i)4-s + (158. + 115. i)5-s + (327. + 238. i)6-s + (−372. + 1.14e3i)7-s + (−142. − 439. i)8-s + (−589. + 428. i)9-s − 2.94e3·10-s + (−3.52e3 + 2.65e3i)11-s − 2.62e3·12-s + (9.80e3 − 7.12e3i)13-s + (−5.59e3 − 1.72e4i)14-s + (1.63e3 − 5.03e3i)15-s + (1.56e4 + 1.13e4i)16-s + (−2.44e4 − 1.77e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.07 + 0.779i)2-s + (−0.178 − 0.549i)3-s + (0.234 − 0.722i)4-s + (0.567 + 0.412i)5-s + (0.619 + 0.450i)6-s + (−0.410 + 1.26i)7-s + (−0.0985 − 0.303i)8-s + (−0.269 + 0.195i)9-s − 0.930·10-s + (−0.799 + 0.601i)11-s − 0.438·12-s + (1.23 − 0.899i)13-s + (−0.544 − 1.67i)14-s + (0.125 − 0.385i)15-s + (0.956 + 0.695i)16-s + (−1.20 − 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.315 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0675160 - 0.0936262i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0675160 - 0.0936262i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.34 + 25.6i)T \) |
| 11 | \( 1 + (3.52e3 - 2.65e3i)T \) |
| good | 2 | \( 1 + (12.1 - 8.82i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-158. - 115. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (372. - 1.14e3i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-9.80e3 + 7.12e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (2.44e4 + 1.77e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (1.09e4 + 3.35e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 7.05e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-2.41e4 + 7.44e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (1.40e5 - 1.02e5i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (4.45e4 - 1.37e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.61e5 + 4.98e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 6.30e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (1.33e5 + 4.10e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (7.86e5 - 5.71e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (3.53e5 - 1.08e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.34e6 + 9.79e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 3.60e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (4.54e5 + 3.30e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-2.57e5 + 7.93e5i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (3.68e6 - 2.67e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-4.87e6 - 3.53e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 6.23e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (2.64e6 - 1.92e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−15.50496833872077552886647193134, −13.54912187831736643377776995005, −12.45554046717061808354603953557, −10.70094230054088590567908415614, −9.314231421283144596483998994959, −8.260979018311293032214224881751, −6.78842180164472962642177969226, −5.77038183414817244616530941077, −2.42955794823218621166057338637, −0.07611338973733041239704299080,
1.59904647170215295067971248777, 3.88494238920210103122252460537, 6.02987985094081993903367143870, 8.230888292831695888335535477293, 9.403923858899141998077390721470, 10.50144347462326681038950940677, 11.13525048300595751330230143908, 13.01793673307310322754640116580, 14.13833559560517572516646077027, 16.09649374223935495140005376963
