| 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
−10.40587857798008499829257019086, −9.816063128138295655956088189562, −8.876443448721047613643032787715, −8.353607439544580634479085851355, −6.56856913953460236259596708531, −5.83761999926101500766632825595, −5.24395119482275089909079725908, −4.20890931893200860331754284782, −2.52594810520528265064159989054, −0.963656835470027132514618518865,
2.09128329650650142012554084581, 3.03499112394836886730767256842, 4.43207165279330226898689217676, 5.22184771051715697808534689293, 6.51492246494510950018513013651, 7.25043767515156858900485485040, 8.389514091744201085165475838202, 9.493989793179848933087376718655, 10.04652562727554336091398007134, 10.93554947286717724484268217917
