| L(s) = 1 | + (2.13 + 1.67i)2-s + (−0.814 − 0.580i)3-s + (1.26 + 5.20i)4-s + (−1.64 + 0.316i)5-s + (−0.763 − 2.60i)6-s + (−2.28 + 1.33i)7-s + (−3.77 + 8.27i)8-s + (0.327 + 0.945i)9-s + (−4.02 − 2.07i)10-s + (0.286 + 0.364i)11-s + (1.98 − 4.96i)12-s + (−0.227 − 0.353i)13-s + (−7.10 − 0.989i)14-s + (1.52 + 0.694i)15-s + (−12.3 + 6.39i)16-s + (1.66 + 1.58i)17-s + ⋯ |
| L(s) = 1 | + (1.50 + 1.18i)2-s + (−0.470 − 0.334i)3-s + (0.630 + 2.60i)4-s + (−0.734 + 0.141i)5-s + (−0.311 − 1.06i)6-s + (−0.863 + 0.503i)7-s + (−1.33 + 2.92i)8-s + (0.109 + 0.315i)9-s + (−1.27 − 0.656i)10-s + (0.0864 + 0.109i)11-s + (0.574 − 1.43i)12-s + (−0.0630 − 0.0980i)13-s + (−1.89 − 0.264i)14-s + (0.392 + 0.179i)15-s + (−3.09 + 1.59i)16-s + (0.403 + 0.384i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.318312 + 2.02083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.318312 + 2.02083i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.814 + 0.580i)T \) |
| 7 | \( 1 + (2.28 - 1.33i)T \) |
| 23 | \( 1 + (-3.01 - 3.72i)T \) |
| good | 2 | \( 1 + (-2.13 - 1.67i)T + (0.471 + 1.94i)T^{2} \) |
| 5 | \( 1 + (1.64 - 0.316i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (-0.286 - 0.364i)T + (-2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (0.227 + 0.353i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.66 - 1.58i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-4.43 + 4.23i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-0.712 + 0.209i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.953 - 9.98i)T + (-30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (-4.98 + 1.72i)T + (29.0 - 22.8i)T^{2} \) |
| 41 | \( 1 + (-1.07 + 0.929i)T + (5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.83 + 1.29i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (4.22 + 2.43i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-12.5 - 0.598i)T + (52.7 + 5.03i)T^{2} \) |
| 59 | \( 1 + (0.205 - 0.398i)T + (-34.2 - 48.0i)T^{2} \) |
| 61 | \( 1 + (6.04 + 8.49i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (4.48 + 11.1i)T + (-48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (1.93 - 13.4i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.84 + 1.66i)T + (64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (-8.01 + 0.381i)T + (78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (5.85 - 6.75i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-12.1 - 1.16i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (8.83 + 10.1i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−11.92938613174896720944317888152, −10.97958089920799642061509500691, −9.346476162798638518502382888519, −8.187095807161111948154457529271, −7.27233679165862530828664580073, −6.74755520685090107579139344005, −5.71484585245943615319776806127, −5.01736890695881667120907252505, −3.74941990827351411859431826995, −2.90520641109736199205716238728,
0.834483612937485936970653427584, 2.81553168928555762920524183932, 3.80800381548371172591474577270, 4.42721625107171427194930847186, 5.59611556544140874834846684723, 6.37354418380459676219305869142, 7.57637367494903933405306237925, 9.421457259619479497362656429926, 10.04977178796610575352307989278, 10.84753313058961652889707097442
