| L(s) = 1 | + (1.85 − 1.19i)2-s + (0.357 − 2.48i)3-s + (1.18 − 2.59i)4-s + (1.18 − 0.348i)5-s + (−2.30 − 5.03i)6-s + (1.55 + 1.79i)7-s + (−0.268 − 1.86i)8-s + (−3.18 − 0.934i)9-s + (1.78 − 2.05i)10-s + (−1.60 − 1.03i)11-s + (−6.03 − 3.88i)12-s + (−0.128 + 0.148i)13-s + (5.01 + 1.47i)14-s + (−0.442 − 3.07i)15-s + (1.01 + 1.17i)16-s + (−0.648 − 1.41i)17-s + ⋯ |
| L(s) = 1 | + (1.31 − 0.842i)2-s + (0.206 − 1.43i)3-s + (0.593 − 1.29i)4-s + (0.530 − 0.155i)5-s + (−0.939 − 2.05i)6-s + (0.587 + 0.677i)7-s + (−0.0949 − 0.660i)8-s + (−1.06 − 0.311i)9-s + (0.564 − 0.651i)10-s + (−0.484 − 0.311i)11-s + (−1.74 − 1.12i)12-s + (−0.0356 + 0.0411i)13-s + (1.34 + 0.393i)14-s + (−0.114 − 0.794i)15-s + (0.254 + 0.293i)16-s + (−0.157 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46863 - 2.95554i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46863 - 2.95554i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (-1.85 + 1.19i)T + (0.830 - 1.81i)T^{2} \) |
| 3 | \( 1 + (-0.357 + 2.48i)T + (-2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (-1.18 + 0.348i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-1.55 - 1.79i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (1.60 + 1.03i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.128 - 0.148i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.648 + 1.41i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (3.31 - 7.26i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.06 - 4.52i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.253 + 1.76i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (3.72 + 1.09i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (0.409 - 0.120i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (0.633 - 4.40i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + (6.43 + 7.42i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.72 + 5.45i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.05 + 7.35i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-6.11 + 3.92i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (0.614 - 0.394i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (2.67 - 5.86i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-3.71 + 4.29i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-12.3 - 3.63i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.97 + 13.7i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (4.14 - 1.21i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.98700103866989849797024359509, −9.913565043667627091071014716638, −8.498474028052429220995707110912, −7.894308809084934013622823197474, −6.52327914811979568477731846405, −5.75009159929404164949129671150, −4.95081685089000973885053852047, −3.45435362792228517202500603836, −2.20998025602574878140814643075, −1.66230650091446981943082345467,
2.65551387505525891919509504693, 3.94848944562163171910107085683, 4.58360232123839962576399684674, 5.24045606445993937720092862477, 6.34505148843628118860584695218, 7.30373453268220339645510287490, 8.400491948801939835303277137316, 9.505643403500595919296929098449, 10.37331494319230922876316632423, 10.96892252010160373779922963063
