| 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
−10.92590536815988942117936320019, −10.31458106645244951786535165187, −9.294362705020545707737110875259, −8.323661732142777805534234566881, −7.19209251900082848960625368988, −6.51989374006952937048843503512, −5.06358300512279964802299636123, −4.36466191371851244366545383326, −2.97805685266290669387694151571, −1.67938068103055734163380366544,
0.26668203519277123869970681517, 2.84262928721175178287391863774, 4.12745219297976996260148139930, 5.12693232185230938431522708586, 5.82610533278342728660266729987, 6.97391390163416984017431090446, 7.83578960646714309082080953021, 8.672283297394173283222380871670, 9.721788403291008986111359530030, 10.41686448797449476108418760171
