L(s) = 1 | + (−2.03 − 0.233i)2-s + (0.653 + 0.475i)3-s + (2.15 + 0.500i)4-s + (−0.564 + 0.825i)5-s + (−1.22 − 1.12i)6-s + (−0.0770 − 0.0792i)7-s + (−0.403 − 0.144i)8-s + (−0.725 − 2.23i)9-s + (1.34 − 1.55i)10-s + (−3.07 + 1.24i)11-s + (1.16 + 1.34i)12-s + (0.978 − 1.62i)13-s + (0.138 + 0.179i)14-s + (−0.761 + 0.271i)15-s + (−3.17 − 1.56i)16-s + (−2.66 + 2.17i)17-s + ⋯ |
L(s) = 1 | + (−1.44 − 0.165i)2-s + (0.377 + 0.274i)3-s + (1.07 + 0.250i)4-s + (−0.252 + 0.369i)5-s + (−0.498 − 0.457i)6-s + (−0.0291 − 0.0299i)7-s + (−0.142 − 0.0509i)8-s + (−0.241 − 0.743i)9-s + (0.424 − 0.490i)10-s + (−0.926 + 0.375i)11-s + (0.337 + 0.389i)12-s + (0.271 − 0.450i)13-s + (0.0370 + 0.0480i)14-s + (−0.196 + 0.0701i)15-s + (−0.793 − 0.390i)16-s + (−0.646 + 0.528i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.118234 - 0.245077i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.118234 - 0.245077i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.564 - 0.825i)T \) |
| 11 | \( 1 + (3.07 - 1.24i)T \) |
good | 2 | \( 1 + (2.03 + 0.233i)T + (1.94 + 0.452i)T^{2} \) |
| 3 | \( 1 + (-0.653 - 0.475i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (0.0770 + 0.0792i)T + (-0.199 + 6.99i)T^{2} \) |
| 13 | \( 1 + (-0.978 + 1.62i)T + (-6.06 - 11.4i)T^{2} \) |
| 17 | \( 1 + (2.66 - 2.17i)T + (3.37 - 16.6i)T^{2} \) |
| 19 | \( 1 + (0.367 + 0.0209i)T + (18.8 + 2.16i)T^{2} \) |
| 23 | \( 1 + (-0.539 + 3.75i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (0.153 - 0.394i)T + (-21.3 - 19.6i)T^{2} \) |
| 31 | \( 1 + (4.52 + 1.91i)T + (21.6 + 22.2i)T^{2} \) |
| 37 | \( 1 + (-0.422 + 4.91i)T + (-36.4 - 6.30i)T^{2} \) |
| 41 | \( 1 + (7.16 - 4.04i)T + (21.1 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-6.37 + 1.87i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (-0.457 + 2.25i)T + (-43.2 - 18.2i)T^{2} \) |
| 53 | \( 1 + (-1.70 + 0.837i)T + (32.3 - 41.9i)T^{2} \) |
| 59 | \( 1 + (6.63 + 3.74i)T + (30.4 + 50.5i)T^{2} \) |
| 61 | \( 1 + (12.3 - 1.41i)T + (59.4 - 13.8i)T^{2} \) |
| 67 | \( 1 + (-0.779 + 1.70i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (2.13 + 8.11i)T + (-61.8 + 34.9i)T^{2} \) |
| 73 | \( 1 + (4.29 + 8.13i)T + (-41.2 + 60.2i)T^{2} \) |
| 79 | \( 1 + (0.414 + 14.5i)T + (-78.8 + 4.51i)T^{2} \) |
| 83 | \( 1 + (14.3 + 2.48i)T + (78.1 + 27.8i)T^{2} \) |
| 89 | \( 1 + (-2.96 - 1.90i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (1.19 + 1.74i)T + (-35.1 + 90.3i)T^{2} \) |
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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
−10.39196178545471423233259036846, −9.360958206562910972350789720402, −8.715677227482518171667046966636, −7.956568357124995368371392631346, −7.14265373349996330834573879821, −6.08042808141773964578703657134, −4.58399614544030114172547915584, −3.27612327390942612063519532316, −2.10630764256176718970229518978, −0.22801783895591577812290676940,
1.54256074293710254127933387171, 2.80029179681859289750265554613, 4.47718980053716553822395203095, 5.64211491839765278884836440653, 7.03832352753256880490672326092, 7.65384543149233299880990734304, 8.457589085339632549358263640286, 8.969780312670581830891728979914, 9.893128792972502465967080932353, 10.89574550047818139614995780070