| L(s) = 1 | + (−0.104 + 0.994i)2-s + (−1.61 + 0.628i)3-s + (−0.978 − 0.207i)4-s + 0.886·5-s + (−0.456 − 1.67i)6-s + (0.891 + 2.74i)7-s + (0.309 − 0.951i)8-s + (2.20 − 2.02i)9-s + (−0.0926 + 0.881i)10-s + (4.04 + 0.860i)11-s + (1.70 − 0.279i)12-s + (3.61 − 2.62i)13-s + (−2.82 + 0.599i)14-s + (−1.43 + 0.557i)15-s + (0.913 + 0.406i)16-s + (−0.966 − 1.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (−0.931 + 0.362i)3-s + (−0.489 − 0.103i)4-s + 0.396·5-s + (−0.186 − 0.682i)6-s + (0.336 + 1.03i)7-s + (0.109 − 0.336i)8-s + (0.736 − 0.676i)9-s + (−0.0292 + 0.278i)10-s + (1.22 + 0.259i)11-s + (0.493 − 0.0806i)12-s + (1.00 − 0.727i)13-s + (−0.753 + 0.160i)14-s + (−0.369 + 0.143i)15-s + (0.228 + 0.101i)16-s + (−0.234 − 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 558 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.614788 + 0.954888i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.614788 + 0.954888i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (1.61 - 0.628i)T \) |
| 31 | \( 1 + (-4.26 - 3.58i)T \) |
| good | 5 | \( 1 - 0.886T + 5T^{2} \) |
| 7 | \( 1 + (-0.891 - 2.74i)T + (-5.66 + 4.11i)T^{2} \) |
| 11 | \( 1 + (-4.04 - 0.860i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.61 + 2.62i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.966 + 1.07i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (4.44 - 1.97i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.80 - 3.11i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (1.09 - 10.4i)T + (-28.3 - 6.02i)T^{2} \) |
| 37 | \( 1 + (-1.98 + 3.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.779 - 0.566i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (5.61 + 4.07i)T + (13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-5.01 - 2.23i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (5.58 - 1.18i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-11.2 - 5.02i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (1.77 + 3.07i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + (2.21 - 0.470i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (5.33 - 5.92i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-3.86 + 11.9i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-7.27 + 3.23i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.22 - 6.83i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.71 - 1.90i)T + (-10.1 - 96.4i)T^{2} \) |
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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
−11.00193157831917446219578735129, −10.15539465631594640759629798129, −9.085864631870965445687490119790, −8.661404791355751435184944369654, −7.22313953474065367568760886293, −6.24714688830782921773372440790, −5.72337432484190557903412694296, −4.79017529790283865481410016119, −3.60568749671419361991092322141, −1.48944889639209224846954795878,
0.873460286222872982687317742736, 1.99274877837040737168298303773, 4.01727371997422630604642310316, 4.49614479371237102298897241953, 6.12774355608658597541752453773, 6.56899341765835210570234264194, 7.83772979595205335167817695499, 8.865877884769134779762012855618, 9.899515365347638755301555004376, 10.66237573230816808817241510329
