L(s) = 1 | + (−0.866 − 0.5i)2-s + (−1.28 − 0.741i)3-s + (0.499 + 0.866i)4-s + (0.977 − 2.01i)5-s + (0.741 + 1.28i)6-s + 0.482i·7-s − 0.999i·8-s + (−0.400 − 0.693i)9-s + (−1.85 + 1.25i)10-s − 4.43·11-s − 1.48i·12-s + (3.58 − 2.07i)13-s + (0.241 − 0.418i)14-s + (−2.74 + 1.85i)15-s + (−0.5 + 0.866i)16-s + (−6.84 − 3.94i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−0.741 − 0.428i)3-s + (0.249 + 0.433i)4-s + (0.437 − 0.899i)5-s + (0.302 + 0.524i)6-s + 0.182i·7-s − 0.353i·8-s + (−0.133 − 0.231i)9-s + (−0.585 + 0.396i)10-s − 1.33·11-s − 0.428i·12-s + (0.994 − 0.574i)13-s + (0.0645 − 0.111i)14-s + (−0.709 + 0.479i)15-s + (−0.125 + 0.216i)16-s + (−1.65 − 0.957i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.183413 - 0.525270i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.183413 - 0.525270i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.977 + 2.01i)T \) |
| 19 | \( 1 + (4.31 + 0.590i)T \) |
good | 3 | \( 1 + (1.28 + 0.741i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 0.482iT - 7T^{2} \) |
| 11 | \( 1 + 4.43T + 11T^{2} \) |
| 13 | \( 1 + (-3.58 + 2.07i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (6.84 + 3.94i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.90 + 2.82i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.91 - 3.31i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.58T + 31T^{2} \) |
| 37 | \( 1 - 3.50iT - 37T^{2} \) |
| 41 | \( 1 + (-4.44 + 7.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.551 - 0.318i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.69 + 2.13i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.76 + 1.59i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.812 + 1.40i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.735 + 1.27i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.4 - 6.59i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.41 - 2.44i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.95 - 3.43i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.576 - 0.997i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 + (-4.37 - 7.58i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.825 + 0.476i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.17567465821597220487059992236, −11.09286822767554026935456947462, −10.41739961535172257369195087763, −8.953536561771649428112648517108, −8.482425717584522175299977580635, −6.94969588207313280588370745175, −5.87594259146610525927218174511, −4.71896344374739652737322902748, −2.56118890833760863771194114275, −0.63770172990894976636477563180,
2.39465502709123643537571496622, 4.46618402484729290974854958671, 5.89128994274755483673654066507, 6.54318393182254308289772721199, 7.88130751476420496119774273949, 8.962093125441480449189922338924, 10.36184510123556567972666298606, 10.74048342020507322434851215401, 11.39296318638623747550444039168, 13.15572838922252028488029787980