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 − 2.64·7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)10-s − 6.29·11-s − 0.999·12-s + (−1.32 + 2.29i)14-s + (−0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (2.82 − 4.88i)17-s − 0.999·18-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 − 0.999·7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 − 0.273i)10-s − 1.89·11-s − 0.288·12-s + (−0.353 + 0.612i)14-s + (−0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (0.684 − 1.18i)17-s − 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0495143 - 1.17540i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0495143 - 1.17540i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.67 + 4.02i)T \) |
good | 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 + 6.29T + 11T^{2} \) |
| 13 | \( 1 + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.82 + 4.88i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.82 - 3.15i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.29T + 31T^{2} \) |
| 37 | \( 1 - 8.29T + 37T^{2} \) |
| 41 | \( 1 + (-4.32 + 7.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3 - 5.19i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.29 + 10.8i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.32 - 4.02i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.64 - 4.58i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.46 + 6.00i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.82 - 8.35i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.82 + 11.8i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.82 - 11.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.64 - 4.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 + (6.61 + 11.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.822 + 1.42i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.21167479761409603935958184887, −9.653050966460284776950613624070, −8.633033013652491677984458721425, −7.66543134650304003649107453855, −6.62687841902815410737963705239, −5.54035984744261608199304680853, −4.68316212317298898606494462527, −3.07140812508194113775194924889, −2.48870576477034165880532921844, −0.54406922090901035924224570178,
2.60320832183567795665031102675, 3.47258175047305593146733615536, 4.66894962608126442547766659051, 5.83051844556322620255475760146, 6.38030463779941462824574333830, 7.88515414424776816259659230251, 8.116640656768927225030350494534, 9.632305621539471641616258867165, 10.11332518572573204153771165085, 10.89722673123411505526782246117