L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s − i·6-s + (0.781 + 2.52i)7-s + (−0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + (−1.31 + 2.27i)11-s + (−0.258 + 0.965i)12-s + (1.21 − 1.21i)13-s + (−0.101 − 2.64i)14-s + (0.500 + 0.866i)16-s + (−7.31 + 1.95i)17-s + (0.965 − 0.258i)18-s + (2.32 + 4.02i)19-s + ⋯ |
L(s) = 1 | + (−0.683 − 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s − 0.408i·6-s + (0.295 + 0.955i)7-s + (−0.249 − 0.249i)8-s + (−0.288 + 0.166i)9-s + (−0.395 + 0.685i)11-s + (−0.0747 + 0.278i)12-s + (0.337 − 0.337i)13-s + (−0.0270 − 0.706i)14-s + (0.125 + 0.216i)16-s + (−1.77 + 0.475i)17-s + (0.227 − 0.0610i)18-s + (0.533 + 0.924i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7481869452\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7481869452\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.781 - 2.52i)T \) |
good | 11 | \( 1 + (1.31 - 2.27i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.21 + 1.21i)T - 13iT^{2} \) |
| 17 | \( 1 + (7.31 - 1.95i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.32 - 4.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.32 + 4.95i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.99iT - 29T^{2} \) |
| 31 | \( 1 + (8.66 + 5.00i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.82 - 1.02i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 5.59iT - 41T^{2} \) |
| 43 | \( 1 + (-0.545 - 0.545i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.64 + 6.12i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (8.28 - 2.22i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.86 + 6.68i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.16 + 2.40i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.663 - 2.47i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 8.36T + 71T^{2} \) |
| 73 | \( 1 + (-3.53 - 13.1i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (7.78 - 4.49i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (7.99 - 7.99i)T - 83iT^{2} \) |
| 89 | \( 1 + (-0.0812 - 0.140i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.35 + 4.35i)T + 97iT^{2} \) |
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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
−10.14368865791495371584782926487, −9.390956334161617379589847446577, −8.656639437059792237412405863338, −8.110267712440986829719676434550, −7.01955334608731691611727693042, −6.00575595200914906559200232861, −5.06359371660029204242715009379, −4.02040479242790185999501227188, −2.75516279887291328671612913181, −1.84234617764990301648166746503,
0.39607434801236351268424718123, 1.72693544441619297898427583403, 2.97998405061657319588125039693, 4.26062760383039950280577114736, 5.42506155072525731276465894343, 6.50485075955325856305383260746, 7.22678605910690465036460105401, 7.78099115571715690765996973180, 8.860189466209288228621650742714, 9.265964110442009885540769617884