L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.384 − 2.20i)5-s + (−0.499 + 0.866i)6-s − 2.51i·7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (0.768 + 2.09i)10-s + 2.88·11-s − 0.999i·12-s + (−4.03 − 2.32i)13-s + (1.25 + 2.18i)14-s + (−0.768 − 2.09i)15-s + (−0.5 − 0.866i)16-s + (−6.31 + 3.64i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.171 − 0.985i)5-s + (−0.204 + 0.353i)6-s − 0.951i·7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.242 + 0.664i)10-s + 0.869·11-s − 0.288i·12-s + (−1.11 − 0.645i)13-s + (0.336 + 0.582i)14-s + (−0.198 − 0.542i)15-s + (−0.125 − 0.216i)16-s + (−1.53 + 0.883i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0490 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0490 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.803362 - 0.843742i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.803362 - 0.843742i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.384 + 2.20i)T \) |
| 19 | \( 1 + (-4.11 + 1.43i)T \) |
good | 7 | \( 1 + 2.51iT - 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 + (4.03 + 2.32i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (6.31 - 3.64i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.48 - 0.855i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.20 + 3.82i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 9.22T + 31T^{2} \) |
| 37 | \( 1 + 3.26iT - 37T^{2} \) |
| 41 | \( 1 + (-4.48 - 7.76i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.84 + 3.37i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.98 + 2.29i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.38 - 5.41i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.73 - 8.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.50 + 12.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.23 + 5.32i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (5.38 + 9.31i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.2 + 7.08i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.11 - 7.13i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7.14iT - 83T^{2} \) |
| 89 | \( 1 + (-2.37 + 4.10i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.98 + 1.14i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.27658006379503282969128852682, −9.347695781805681976392198361627, −8.891158512320538397548870003001, −7.78625885180554925434200684643, −7.21003057808268470914819872178, −6.13444667749638940970772919942, −4.89289027342060229453143336795, −3.87199006726317981046084557811, −2.10513638000661952733863513143, −0.75813841244013705124521789867,
2.09256242134810151151352496589, 2.81937236431183300766929127523, 4.07127143491831289002077982874, 5.44537707796930227491984416567, 6.86202794678397991374763581225, 7.26135948564847193509056221384, 8.693313699171519054096992045185, 9.277356349159915422785488432551, 9.850881145578584332533718039361, 10.98186774610712773864430558967