| L(s) = 1 | + (−2.10 − 2.90i)2-s + (0.307 − 2.98i)3-s + (−2.74 + 8.43i)4-s + (−1.22 + 1.68i)5-s + (−9.30 + 5.40i)6-s + (2.73 − 8.42i)7-s + (16.6 − 5.39i)8-s + (−8.81 − 1.83i)9-s + 7.48·10-s + (10.8 − 2.03i)11-s + (24.3 + 10.7i)12-s + (1.33 − 0.966i)13-s + (−30.2 + 9.82i)14-s + (4.66 + 4.17i)15-s + (−21.9 − 15.9i)16-s + (7.30 − 10.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 1.45i)2-s + (0.102 − 0.994i)3-s + (−0.685 + 2.10i)4-s + (−0.245 + 0.337i)5-s + (−1.55 + 0.900i)6-s + (0.391 − 1.20i)7-s + (2.07 − 0.674i)8-s + (−0.979 − 0.203i)9-s + 0.748·10-s + (0.982 − 0.184i)11-s + (2.02 + 0.897i)12-s + (0.102 − 0.0743i)13-s + (−2.15 + 0.701i)14-s + (0.310 + 0.278i)15-s + (−1.37 − 0.998i)16-s + (0.429 − 0.591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.113196 - 0.598919i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113196 - 0.598919i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.307 + 2.98i)T \) |
| 11 | \( 1 + (-10.8 + 2.03i)T \) |
| good | 2 | \( 1 + (2.10 + 2.90i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (1.22 - 1.68i)T + (-7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (-2.73 + 8.42i)T + (-39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (-1.33 + 0.966i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (-7.30 + 10.0i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (-3.26 - 10.0i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 - 20.3iT - 529T^{2} \) |
| 29 | \( 1 + (-11.0 - 3.58i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-18.9 + 13.7i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-2.23 + 6.89i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (36.9 - 12.0i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + 15.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (43.0 - 14.0i)T + (1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (23.6 + 32.6i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-107. - 34.9i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (62.7 + 45.5i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 - 62.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + (6.00 - 8.26i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (23.0 - 70.9i)T + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (69.8 - 50.7i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-6.94 + 9.55i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 - 74.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-62.4 + 45.3i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−16.80346315486913286745376384497, −14.29114886895562893062432916088, −13.22258158376487184021283908376, −11.83515366966605111912495258267, −11.17306446186757613059986262047, −9.752996849233461148815056507584, −8.246231499197666691521118540011, −7.15305199529400400114376560747, −3.47296378048369802836646352750, −1.23799141357952671671960465085,
4.89829003821551526995068207709, 6.30873023898686100171081878142, 8.353624370159210556586550458578, 8.928896903220845222880700699701, 10.16457075835514273187221773530, 11.86360220786173054046470224702, 14.33207459793643408955831779352, 15.06192447414422731342494720365, 15.92627279289833571824288883369, 16.82407829277319123258608011778
