| L(s) = 1 | + (0.886 + 0.322i)2-s + (−0.292 + 1.70i)3-s + (−0.850 − 0.713i)4-s + (0.485 + 0.578i)5-s + (−0.809 + 1.41i)6-s + (1.38 − 2.39i)7-s + (−1.46 − 2.54i)8-s + (−2.82 − 0.997i)9-s + (0.243 + 0.669i)10-s + (−2.28 + 1.31i)11-s + (1.46 − 1.24i)12-s + (1.90 + 0.335i)13-s + (1.99 − 1.67i)14-s + (−1.13 + 0.660i)15-s + (−0.0947 − 0.537i)16-s + (0.220 − 0.606i)17-s + ⋯ |
| L(s) = 1 | + (0.626 + 0.228i)2-s + (−0.168 + 0.985i)3-s + (−0.425 − 0.356i)4-s + (0.217 + 0.258i)5-s + (−0.330 + 0.579i)6-s + (0.522 − 0.905i)7-s + (−0.518 − 0.898i)8-s + (−0.943 − 0.332i)9-s + (0.0770 + 0.211i)10-s + (−0.687 + 0.396i)11-s + (0.423 − 0.359i)12-s + (0.527 + 0.0930i)13-s + (0.534 − 0.448i)14-s + (−0.291 + 0.170i)15-s + (−0.0236 − 0.134i)16-s + (0.0535 − 0.147i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.799 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.950776 + 0.317689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.950776 + 0.317689i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.292 - 1.70i)T \) |
| 19 | \( 1 + (1.94 - 3.90i)T \) |
| good | 2 | \( 1 + (-0.886 - 0.322i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.485 - 0.578i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.38 + 2.39i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.28 - 1.31i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.90 - 0.335i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.220 + 0.606i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (5.61 - 6.69i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-8.33 + 3.03i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.10 - 1.21i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.09iT - 37T^{2} \) |
| 41 | \( 1 + (0.930 + 5.27i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1.12 - 0.947i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (3.48 + 9.56i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (4.16 + 3.49i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-7.06 - 2.57i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.37 + 1.15i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.35 + 9.20i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (2.08 - 1.74i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.952 - 5.39i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-9.22 + 1.62i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.02 - 4.05i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.233 + 1.32i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.64 + 12.7i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−15.27469970438643501195450773297, −14.17887535653483333184676965315, −13.62578466929609396123676511143, −11.95787477800527420094023711862, −10.42797705697286736403324964968, −9.940624361556344808872312343225, −8.238999125872871437091250287437, −6.28107073175052526795200706681, −4.96062605106334285854110791589, −3.84256181240069650223152206107,
2.61813033520753190504151594410, 4.93999522531040854838516867441, 6.12635160791796877658300468145, 8.107888411161802633793631612891, 8.784712059537361134272356116189, 10.99420061912256143056176155470, 12.10109376327517736420541434099, 12.86051265833739446537669908417, 13.73235697552766500059522601260, 14.73362286852160524584237218640
