L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.20 + 2.09i)5-s − 0.999·6-s + (1.62 − 2.09i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.20 + 2.09i)10-s + (−1 + 1.73i)11-s + (−0.499 − 0.866i)12-s + 3.82·13-s + (2.62 + 0.358i)14-s − 2.41·15-s + (−0.5 − 0.866i)16-s + (−2.62 + 4.54i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.539 + 0.935i)5-s − 0.408·6-s + (0.612 − 0.790i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.381 + 0.661i)10-s + (−0.301 + 0.522i)11-s + (−0.144 − 0.249i)12-s + 1.06·13-s + (0.700 + 0.0958i)14-s − 0.623·15-s + (−0.125 − 0.216i)16-s + (−0.635 + 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.660071 + 1.74574i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.660071 + 1.74574i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.62 + 2.09i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.20 - 2.09i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + (2.62 - 4.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.82 - 6.63i)T + (-9.5 + 16.4i)T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.41 - 7.64i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.03 - 3.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.82 + 8.36i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.44 + 4.24i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 + (0.328 - 0.568i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.17T + 83T^{2} \) |
| 89 | \( 1 + (3.41 + 5.91i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.65T + 97T^{2} \) |
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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
−10.31042525341625796900987277070, −9.756720790930085387774841241217, −8.434141771873790579084160042281, −7.78792977970786782440009198286, −6.73446252760113160254150268192, −6.11987185532780520732701073903, −5.22342278237550860600683172908, −4.11440732216621323316351872479, −3.42827224746497943516543998416, −1.77450182969700944211709697250,
0.843671661703862099221753662900, 1.96733475170976023482029836800, 3.05149943405001832619353397817, 4.60132355544210170308474712470, 5.33398415865228400730036849986, 5.87284059447205433596053655789, 7.08808798045218545249867190406, 8.247222606922866574968626177651, 9.071472616825149564512247079589, 9.436291279806618062324934527495