| L(s) = 1 | + (0.760 + 0.161i)2-s + (−0.0646 + 0.614i)3-s + (−1.27 − 0.567i)4-s + (−0.148 − 0.165i)5-s + (−0.148 + 0.456i)6-s + (−2.13 − 1.55i)8-s + (2.56 + 0.544i)9-s + (−0.0865 − 0.149i)10-s + (2.48 − 2.19i)11-s + (0.431 − 0.746i)12-s + (2.01 + 6.20i)13-s + (0.111 − 0.0808i)15-s + (0.494 + 0.548i)16-s + (4.23 − 0.901i)17-s + (1.85 + 0.827i)18-s + (2.66 − 1.18i)19-s + ⋯ |
| L(s) = 1 | + (0.537 + 0.114i)2-s + (−0.0372 + 0.354i)3-s + (−0.637 − 0.283i)4-s + (−0.0665 − 0.0739i)5-s + (−0.0606 + 0.186i)6-s + (−0.755 − 0.548i)8-s + (0.853 + 0.181i)9-s + (−0.0273 − 0.0473i)10-s + (0.749 − 0.662i)11-s + (0.124 − 0.215i)12-s + (0.559 + 1.72i)13-s + (0.0287 − 0.0208i)15-s + (0.123 + 0.137i)16-s + (1.02 − 0.218i)17-s + (0.438 + 0.195i)18-s + (0.610 − 0.272i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.73532 + 0.206980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73532 + 0.206980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-2.48 + 2.19i)T \) |
| good | 2 | \( 1 + (-0.760 - 0.161i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (0.0646 - 0.614i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (0.148 + 0.165i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (-2.01 - 6.20i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-4.23 + 0.901i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (-2.66 + 1.18i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.94 + 3.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.05 + 2.21i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (4.61 - 5.11i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.590 - 5.62i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (1.08 + 0.786i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 + (-5.52 + 2.45i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (1.14 - 1.27i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (8.70 + 3.87i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (6.44 + 7.15i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (0.635 + 1.10i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.87 - 8.85i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-5.10 - 2.27i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-4.43 - 0.941i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-3.48 + 10.7i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (3.96 - 6.86i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.79 + 8.61i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.82477739955000308229140382513, −9.839308738488310885563171275236, −9.193049392295907598052457411788, −8.409240024084329906873102351503, −6.94376138351776854636999420255, −6.22688446806741743033866792289, −4.99492462979983483123251359237, −4.28316887834828480324522718532, −3.39524977488763231650670738575, −1.27461956013695230327407548234,
1.24154912972109292274462110626, 3.19651385873742635645271328599, 3.92354281869559652004212643569, 5.17961166115778552754316169607, 5.95727782044394566785914550625, 7.36329054155801467812960319867, 7.88859651723071866796109659123, 9.132407704900581678917517748397, 9.822016889082447855823324745965, 10.84330511054570639631731188515
