| L(s) = 1 | + (−1.38 + 0.451i)2-s + (1.86 + 2.56i)3-s + (0.109 − 0.0798i)4-s + (2.21 − 0.315i)5-s + (−3.75 − 2.72i)6-s + 0.259i·7-s + (1.60 − 2.20i)8-s + (−2.18 + 6.73i)9-s + (−2.93 + 1.43i)10-s + (0.353 + 1.08i)11-s + (0.410 + 0.133i)12-s + (0.350 + 0.113i)13-s + (−0.116 − 0.359i)14-s + (4.94 + 5.09i)15-s + (−1.31 + 4.04i)16-s + (−0.587 + 0.809i)17-s + ⋯ |
| L(s) = 1 | + (−0.982 + 0.319i)2-s + (1.07 + 1.48i)3-s + (0.0549 − 0.0399i)4-s + (0.990 − 0.140i)5-s + (−1.53 − 1.11i)6-s + 0.0978i·7-s + (0.566 − 0.779i)8-s + (−0.729 + 2.24i)9-s + (−0.927 + 0.454i)10-s + (0.106 + 0.327i)11-s + (0.118 + 0.0384i)12-s + (0.0971 + 0.0315i)13-s + (−0.0312 − 0.0962i)14-s + (1.27 + 1.31i)15-s + (−0.328 + 1.01i)16-s + (−0.142 + 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.563074 + 1.14962i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563074 + 1.14962i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.21 + 0.315i)T \) |
| 17 | \( 1 + (0.587 - 0.809i)T \) |
| good | 2 | \( 1 + (1.38 - 0.451i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.86 - 2.56i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 0.259iT - 7T^{2} \) |
| 11 | \( 1 + (-0.353 - 1.08i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.350 - 0.113i)T + (10.5 + 7.64i)T^{2} \) |
| 19 | \( 1 + (-2.44 - 1.77i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (2.23 - 0.725i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.99 + 4.35i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (5.81 + 4.22i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (9.73 + 3.16i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.123 + 0.380i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.20iT - 43T^{2} \) |
| 47 | \( 1 + (4.59 + 6.32i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.64 - 7.76i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.50 - 7.69i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.665 + 2.04i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.79 + 10.7i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-2.00 + 1.45i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.02 + 0.331i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.43 + 1.03i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (7.49 - 10.3i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.317 - 0.977i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (8.37 + 11.5i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.82953334045112684525178019763, −10.06864954427573601969335683844, −9.660021785016892344961042761285, −8.886763291104225782336826995681, −8.331045343621572329823946462720, −7.23252885450522882466731545430, −5.70291113009426330692946450298, −4.54739179129775362647569013954, −3.56108919785845002234261543290, −2.07604596340520972337360122104,
1.11134242351061331659562331994, 2.06118591941156849650974784926, 3.14495120860400443331242401064, 5.28325514320516731345585766818, 6.57751820587472998100356291308, 7.29991171659960358840586803157, 8.420945763934103549480489836900, 8.860168121545558175004671019387, 9.697847317199886068224777586944, 10.62081971095051825232383849705
