L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.766 + 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.766 − 0.642i)6-s + (−0.284 + 0.493i)7-s + (−0.5 − 0.866i)8-s + (0.173 − 0.984i)9-s + (0.173 − 0.984i)10-s + (−0.953 − 1.65i)11-s + (0.499 − 0.866i)12-s + (−3.59 − 3.02i)13-s + (−0.535 − 0.194i)14-s + (0.939 − 0.342i)15-s + (0.766 − 0.642i)16-s + (−0.808 − 4.58i)17-s + ⋯ |
L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.442 + 0.371i)3-s + (−0.469 + 0.171i)4-s + (−0.420 − 0.152i)5-s + (−0.312 − 0.262i)6-s + (−0.107 + 0.186i)7-s + (−0.176 − 0.306i)8-s + (0.0578 − 0.328i)9-s + (0.0549 − 0.311i)10-s + (−0.287 − 0.497i)11-s + (0.144 − 0.249i)12-s + (−0.998 − 0.837i)13-s + (−0.143 − 0.0520i)14-s + (0.242 − 0.0883i)15-s + (0.191 − 0.160i)16-s + (−0.196 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.383379 - 0.270151i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.383379 - 0.270151i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 + (1.57 - 4.06i)T \) |
good | 7 | \( 1 + (0.284 - 0.493i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.953 + 1.65i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3.59 + 3.02i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.808 + 4.58i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.76 + 0.643i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.782 + 4.43i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.44 + 5.96i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 + (-2.55 + 2.14i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.29 + 2.29i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.88 + 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (7.96 - 2.89i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.205 - 1.16i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (0.352 - 0.128i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.00191 + 0.0108i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.16 - 1.51i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (10.4 - 8.79i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (11.8 - 9.91i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.95 + 6.84i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (12.0 + 10.1i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.389 + 2.21i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.41686448797449476108418760171, −9.721788403291008986111359530030, −8.672283297394173283222380871670, −7.83578960646714309082080953021, −6.97391390163416984017431090446, −5.82610533278342728660266729987, −5.12693232185230938431522708586, −4.12745219297976996260148139930, −2.84262928721175178287391863774, −0.26668203519277123869970681517,
1.67938068103055734163380366544, 2.97805685266290669387694151571, 4.36466191371851244366545383326, 5.06358300512279964802299636123, 6.51989374006952937048843503512, 7.19209251900082848960625368988, 8.323661732142777805534234566881, 9.294362705020545707737110875259, 10.31458106645244951786535165187, 10.92590536815988942117936320019