| L(s) = 1 | + (0.186 − 1.29i)2-s + (0.415 + 0.909i)3-s + (0.273 + 0.0801i)4-s + (1.65 + 1.90i)5-s + (1.25 − 0.368i)6-s + (0.841 + 0.540i)7-s + (1.24 − 2.72i)8-s + (−0.654 + 0.755i)9-s + (2.77 − 1.78i)10-s + (0.470 + 3.27i)11-s + (0.0405 + 0.281i)12-s + (−2.21 + 1.42i)13-s + (0.857 − 0.989i)14-s + (−1.04 + 2.29i)15-s + (−2.81 − 1.81i)16-s + (−2.87 + 0.843i)17-s + ⋯ |
| L(s) = 1 | + (0.131 − 0.916i)2-s + (0.239 + 0.525i)3-s + (0.136 + 0.0400i)4-s + (0.738 + 0.852i)5-s + (0.513 − 0.150i)6-s + (0.317 + 0.204i)7-s + (0.439 − 0.962i)8-s + (−0.218 + 0.251i)9-s + (0.878 − 0.564i)10-s + (0.141 + 0.986i)11-s + (0.0116 + 0.0813i)12-s + (−0.615 + 0.395i)13-s + (0.229 − 0.264i)14-s + (−0.270 + 0.592i)15-s + (−0.704 − 0.452i)16-s + (−0.696 + 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0131i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07502 + 0.0136880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07502 + 0.0136880i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 23 | \( 1 + (-4.63 + 1.24i)T \) |
| good | 2 | \( 1 + (-0.186 + 1.29i)T + (-1.91 - 0.563i)T^{2} \) |
| 5 | \( 1 + (-1.65 - 1.90i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.470 - 3.27i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (2.21 - 1.42i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (2.87 - 0.843i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (4.39 + 1.29i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-8.79 + 2.58i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (0.0842 - 0.184i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (1.42 - 1.63i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (8.17 + 9.43i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.94 + 8.63i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 5.57T + 47T^{2} \) |
| 53 | \( 1 + (-8.67 - 5.57i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.24 + 0.800i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-1.38 + 3.03i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.58 + 11.0i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-1.08 + 7.53i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-4.67 - 1.37i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (14.2 - 9.15i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (3.75 - 4.33i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.48 - 12.0i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (4.55 + 5.26i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.62575509024024573414754814327, −10.46939108887195713075508319550, −9.537838842952437638828251073265, −8.576322384558449647590287134900, −7.04959112165078693888268134538, −6.58561338655097336022482653670, −4.96228394547578409615514780591, −4.01347824809972523090618578441, −2.60282000933952006111123696448, −2.07913971279402450357802337394,
1.36727850234478730778449611855, 2.73332811052003537948259590766, 4.66976109054865085698520687607, 5.48833617109183890219692848179, 6.40567110014006875909810713390, 7.17008357996907649071297184231, 8.425260166276038398046383021823, 8.633815426172076502267935631024, 10.01214448436424253875122874567, 11.03342253051575294403904232342
