L(s) = 1 | + (−1.99 − 1.44i)2-s + (0.5 − 1.53i)3-s + (1.26 + 3.88i)4-s + (−2.80 + 2.03i)5-s + (−3.22 + 2.34i)6-s + (1.58 − 4.89i)8-s + (0.309 + 0.224i)9-s + 8.55·10-s + (2.91 − 1.57i)11-s + 6.60·12-s + (−0.528 − 0.384i)13-s + (1.73 + 5.33i)15-s + (−3.65 + 2.65i)16-s + (−0.919 + 0.668i)17-s + (−0.291 − 0.896i)18-s + (1.87 − 5.77i)19-s + ⋯ |
L(s) = 1 | + (−1.41 − 1.02i)2-s + (0.288 − 0.888i)3-s + (0.631 + 1.94i)4-s + (−1.25 + 0.911i)5-s + (−1.31 + 0.957i)6-s + (0.561 − 1.72i)8-s + (0.103 + 0.0748i)9-s + 2.70·10-s + (0.880 − 0.474i)11-s + 1.90·12-s + (−0.146 − 0.106i)13-s + (0.447 + 1.37i)15-s + (−0.913 + 0.663i)16-s + (−0.223 + 0.162i)17-s + (−0.0686 − 0.211i)18-s + (0.430 − 1.32i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.133565 - 0.504264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.133565 - 0.504264i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 + (-2.91 + 1.57i)T \) |
good | 2 | \( 1 + (1.99 + 1.44i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.5 + 1.53i)T + (-2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (2.80 - 2.03i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (0.528 + 0.384i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (0.919 - 0.668i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.87 + 5.77i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 6.66T + 23T^{2} \) |
| 29 | \( 1 + (1.41 + 4.34i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.26 - 1.64i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.135 + 0.418i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.82 + 5.61i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + (-0.186 + 0.575i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (7.94 + 5.77i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.523 + 1.61i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.54 + 4.03i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 + (-4.38 + 3.18i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.07 + 6.37i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (2.14 + 1.55i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-5.41 + 3.93i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 0.698T + 89T^{2} \) |
| 97 | \( 1 + (12.0 + 8.73i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.60121438725313550896164561448, −9.583053962708694118007678797810, −8.618710764826849181084239634704, −7.84653320293808664556728788403, −7.32849342696292300473312313888, −6.47068419423133800607143305091, −4.13136942357903022238739592667, −3.10719803641808313271114866772, −2.08059041803105408942684592682, −0.53953412143106605936152423069,
1.26133703331609477442461635164, 3.81147481718060232861716148673, 4.54024727259950865541387101673, 5.86937704438441585562968540504, 7.04790936257308529246397009659, 7.87039767096912319438953879240, 8.504160517451934678935395145068, 9.377032808843469192635716019038, 9.778457065369599646495532851773, 10.80044584032641370351653751315