| L(s) = 1 | + (0.554 − 0.256i)2-s + (−1.05 + 1.23i)4-s + (2.81 + 1.69i)5-s + (0.0467 − 0.862i)7-s + (−0.593 + 2.13i)8-s + (1.99 + 0.217i)10-s + (−0.631 + 3.84i)11-s + (−1.31 + 0.998i)13-s + (−0.195 − 0.490i)14-s + (−0.306 − 1.87i)16-s + (−0.182 − 3.36i)17-s + (−3.16 + 2.99i)19-s + (−5.06 + 1.70i)20-s + (0.637 + 2.29i)22-s + (2.04 − 0.450i)23-s + ⋯ |
| L(s) = 1 | + (0.392 − 0.181i)2-s + (−0.526 + 0.619i)4-s + (1.26 + 0.758i)5-s + (0.0176 − 0.325i)7-s + (−0.209 + 0.755i)8-s + (0.632 + 0.0687i)10-s + (−0.190 + 1.16i)11-s + (−0.364 + 0.276i)13-s + (−0.0522 − 0.131i)14-s + (−0.0767 − 0.468i)16-s + (−0.0442 − 0.815i)17-s + (−0.726 + 0.688i)19-s + (−1.13 + 0.381i)20-s + (0.136 + 0.489i)22-s + (0.426 − 0.0938i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42303 + 1.02639i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42303 + 1.02639i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-3.82 + 6.65i)T \) |
| good | 2 | \( 1 + (-0.554 + 0.256i)T + (1.29 - 1.52i)T^{2} \) |
| 5 | \( 1 + (-2.81 - 1.69i)T + (2.34 + 4.41i)T^{2} \) |
| 7 | \( 1 + (-0.0467 + 0.862i)T + (-6.95 - 0.756i)T^{2} \) |
| 11 | \( 1 + (0.631 - 3.84i)T + (-10.4 - 3.51i)T^{2} \) |
| 13 | \( 1 + (1.31 - 0.998i)T + (3.47 - 12.5i)T^{2} \) |
| 17 | \( 1 + (0.182 + 3.36i)T + (-16.9 + 1.83i)T^{2} \) |
| 19 | \( 1 + (3.16 - 2.99i)T + (1.02 - 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.04 + 0.450i)T + (20.8 - 9.65i)T^{2} \) |
| 29 | \( 1 + (-3.93 - 1.82i)T + (18.7 + 22.1i)T^{2} \) |
| 31 | \( 1 + (-5.88 - 5.57i)T + (1.67 + 30.9i)T^{2} \) |
| 37 | \( 1 + (0.331 + 1.19i)T + (-31.7 + 19.0i)T^{2} \) |
| 41 | \( 1 + (-3.25 - 0.716i)T + (37.2 + 17.2i)T^{2} \) |
| 43 | \( 1 + (0.471 + 2.87i)T + (-40.7 + 13.7i)T^{2} \) |
| 47 | \( 1 + (5.86 - 3.52i)T + (22.0 - 41.5i)T^{2} \) |
| 53 | \( 1 + (2.19 - 0.239i)T + (51.7 - 11.3i)T^{2} \) |
| 61 | \( 1 + (-10.6 + 4.92i)T + (39.4 - 46.4i)T^{2} \) |
| 67 | \( 1 + (-1.13 + 4.10i)T + (-57.4 - 34.5i)T^{2} \) |
| 71 | \( 1 + (-9.08 + 5.46i)T + (33.2 - 62.7i)T^{2} \) |
| 73 | \( 1 + (4.53 + 11.3i)T + (-52.9 + 50.2i)T^{2} \) |
| 79 | \( 1 + (9.88 - 3.33i)T + (62.8 - 47.8i)T^{2} \) |
| 83 | \( 1 + (3.22 - 4.76i)T + (-30.7 - 77.1i)T^{2} \) |
| 89 | \( 1 + (2.24 + 1.03i)T + (57.6 + 67.8i)T^{2} \) |
| 97 | \( 1 + (-3.53 + 8.87i)T + (-70.4 - 66.7i)T^{2} \) |
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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
−10.93554947286717724484268217917, −10.04652562727554336091398007134, −9.493989793179848933087376718655, −8.389514091744201085165475838202, −7.25043767515156858900485485040, −6.51492246494510950018513013651, −5.22184771051715697808534689293, −4.43207165279330226898689217676, −3.03499112394836886730767256842, −2.09128329650650142012554084581,
0.963656835470027132514618518865, 2.52594810520528265064159989054, 4.20890931893200860331754284782, 5.24395119482275089909079725908, 5.83761999926101500766632825595, 6.56856913953460236259596708531, 8.353607439544580634479085851355, 8.876443448721047613643032787715, 9.816063128138295655956088189562, 10.40587857798008499829257019086
