L(s) = 1 | + (−1 − i)2-s + (−1.5 − 0.866i)3-s + 2i·4-s + (−0.5 − 0.866i)5-s + (0.633 + 2.36i)6-s + (−0.866 − 2.5i)7-s + (2 − 2i)8-s + (−0.366 + 1.36i)10-s + (−2.73 + 4.73i)11-s + (1.73 − 3i)12-s − 1.26·13-s + (−1.63 + 3.36i)14-s + 1.73i·15-s − 4·16-s + (4.09 + 2.36i)17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.866 − 0.499i)3-s + i·4-s + (−0.223 − 0.387i)5-s + (0.258 + 0.965i)6-s + (−0.327 − 0.944i)7-s + (0.707 − 0.707i)8-s + (−0.115 + 0.431i)10-s + (−0.823 + 1.42i)11-s + (0.499 − 0.866i)12-s − 0.351·13-s + (−0.436 + 0.899i)14-s + 0.447i·15-s − 16-s + (0.993 + 0.573i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 3 | \( 1 + (1.5 + 0.866i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (2.73 - 4.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 + (-4.09 - 2.36i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.90 - 1.09i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.86 - 3.96i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.53iT - 29T^{2} \) |
| 31 | \( 1 + (-4.09 + 7.09i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 9.92T + 43T^{2} \) |
| 47 | \( 1 + (4.73 + 8.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.83 + 5.09i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.09 + 0.633i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.86 + 6.69i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.69 + 8.13i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.73iT - 71T^{2} \) |
| 73 | \( 1 + (4.90 + 2.83i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.09 - 0.633i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.66iT - 83T^{2} \) |
| 89 | \( 1 + (-2.30 + 1.33i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
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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
−11.25704845396704398800159276642, −10.06414072931174097860296277798, −9.803251194063441923130431728283, −8.077371125147485741836240099910, −7.49070242268475714322864488203, −6.40660556810252431110409381198, −4.85780221458947423922527835508, −3.60869616102761753444986059970, −1.68517863689454721011176667104, 0,
2.77494325968640988911933112943, 4.85079335238427475029557264149, 5.76279013121502496092617392208, 6.37421838365332261072671762417, 7.893780494438213361211865683266, 8.555486439842126471054043189548, 9.860681449651809871547997402257, 10.49667058478195244349522643468, 11.36371916758799169089474686085