L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.499 − 0.866i)6-s − 7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.499 − 0.866i)10-s − 4.77·11-s + 0.999·12-s + (−1.88 + 3.26i)13-s + (−0.5 − 0.866i)14-s + (−0.499 + 0.866i)15-s + (−0.5 − 0.866i)16-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.204 − 0.353i)6-s − 0.377·7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s − 1.43·11-s + 0.288·12-s + (−0.523 + 0.906i)13-s + (−0.133 − 0.231i)14-s + (−0.129 + 0.223i)15-s + (−0.125 − 0.216i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0128i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0128i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00179703 + 0.279854i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00179703 + 0.279854i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (1.38 - 4.13i)T \) |
good | 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 4.77T + 11T^{2} \) |
| 13 | \( 1 + (1.88 - 3.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (2.38 - 4.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + (2.38 + 4.13i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.113 + 0.197i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.613 - 1.06i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.77 + 3.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.88 + 4.99i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.11 + 1.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.77 + 8.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.88 - 10.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.65 - 13.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (-4.15 + 7.20i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.37382755973769663419724896702, −10.25184795248686885223432368145, −9.342046680625294717729957414009, −8.160279572286393682402300817129, −7.63486540773088237729388052755, −6.64390014301255089113725834474, −5.68611338796817555120309423138, −4.87892045185958998425650059452, −3.67021370753673792079380701801, −2.14941285181559990841856776590,
0.13528159176385247344993831928, 2.53392617694469497759118158751, 3.32721916202657845987984950242, 4.67675488311901983742177546034, 5.37490950381912048044220657753, 6.50482067623827917410427134437, 7.63692490956924695363476006758, 8.678955849365812355778499503767, 9.796554957946411638354945266163, 10.52957039612211570109706544884