| L(s) = 1 | + (1.04 − 0.818i)2-s + (0.814 − 0.580i)3-s + (−0.0577 + 0.238i)4-s + (−1.14 − 0.220i)5-s + (0.373 − 1.27i)6-s + (1.26 + 2.32i)7-s + (1.23 + 2.70i)8-s + (0.327 − 0.945i)9-s + (−1.37 + 0.707i)10-s + (1.25 − 1.59i)11-s + (0.0910 + 0.227i)12-s + (3.57 − 5.56i)13-s + (3.22 + 1.37i)14-s + (−1.06 + 0.484i)15-s + (3.06 + 1.58i)16-s + (2.95 − 2.81i)17-s + ⋯ |
| L(s) = 1 | + (0.736 − 0.579i)2-s + (0.470 − 0.334i)3-s + (−0.0288 + 0.119i)4-s + (−0.512 − 0.0986i)5-s + (0.152 − 0.518i)6-s + (0.479 + 0.877i)7-s + (0.436 + 0.956i)8-s + (0.109 − 0.315i)9-s + (−0.434 + 0.223i)10-s + (0.379 − 0.481i)11-s + (0.0262 + 0.0656i)12-s + (0.991 − 1.54i)13-s + (0.861 + 0.368i)14-s + (−0.273 + 0.125i)15-s + (0.766 + 0.395i)16-s + (0.715 − 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 + 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.32067 - 0.610749i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32067 - 0.610749i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.26 - 2.32i)T \) |
| 23 | \( 1 + (3.94 - 2.73i)T \) |
| good | 2 | \( 1 + (-1.04 + 0.818i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (1.14 + 0.220i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (-1.25 + 1.59i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.57 + 5.56i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-2.95 + 2.81i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.18 - 4.94i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (4.96 + 1.45i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.777 - 8.14i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (1.43 + 0.495i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (5.03 + 4.35i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.80 + 1.27i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (5.85 - 3.38i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.63 + 0.0780i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (-1.95 - 3.79i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-1.82 + 2.56i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-2.21 + 5.52i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (1.97 + 13.7i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (3.97 + 0.964i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-4.37 - 0.208i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (5.81 + 6.71i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-17.3 + 1.65i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-3.27 + 3.78i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.31504094846209240493138403350, −10.18529930691827977638823670541, −8.918921297187059047380104700965, −8.033756899328338559375411765646, −7.71918451277411493611754442282, −5.85572595786862237433258253347, −5.21210513513948573833613040328, −3.58897422794763205854663953902, −3.23480085395732978362797829748, −1.63663958116280494748020030393,
1.54373107465826656384920535299, 3.81104775208671903602975459150, 4.10114290576455263246091595114, 5.22657481229892030194942363572, 6.50294603000972691942365166388, 7.25543900885792756242820610326, 8.139567111650068203453285127737, 9.380593473614467712650678412340, 10.03768769588227864195908656939, 11.18772557401156147960997670766
