| L(s) = 1 | + (−0.642 − 0.766i)2-s + (1.61 + 0.638i)3-s + (−0.173 + 0.984i)4-s + (1.09 + 1.95i)5-s + (−0.546 − 1.64i)6-s + (1.53 + 0.888i)7-s + (0.866 − 0.500i)8-s + (2.18 + 2.05i)9-s + (0.791 − 2.09i)10-s + (−3.70 + 2.14i)11-s + (−0.908 + 1.47i)12-s + (2.04 − 0.743i)13-s + (−0.308 − 1.75i)14-s + (0.514 + 3.83i)15-s + (−0.939 − 0.342i)16-s + (−0.583 + 0.489i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (0.929 + 0.368i)3-s + (−0.0868 + 0.492i)4-s + (0.488 + 0.872i)5-s + (−0.222 − 0.671i)6-s + (0.581 + 0.335i)7-s + (0.306 − 0.176i)8-s + (0.728 + 0.685i)9-s + (0.250 − 0.661i)10-s + (−1.11 + 0.645i)11-s + (−0.262 + 0.425i)12-s + (0.566 − 0.206i)13-s + (−0.0824 − 0.467i)14-s + (0.132 + 0.991i)15-s + (−0.234 − 0.0855i)16-s + (−0.141 + 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57827 + 0.644109i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57827 + 0.644109i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 + 0.766i)T \) |
| 3 | \( 1 + (-1.61 - 0.638i)T \) |
| 5 | \( 1 + (-1.09 - 1.95i)T \) |
| 19 | \( 1 + (4.17 - 1.26i)T \) |
| good | 7 | \( 1 + (-1.53 - 0.888i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.70 - 2.14i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.04 + 0.743i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.583 - 0.489i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.10 + 6.28i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.06 - 0.894i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (4.34 + 2.50i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.40T + 37T^{2} \) |
| 41 | \( 1 + (-3.10 - 1.13i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.00915 + 0.00161i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.61 - 3.03i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.204 - 0.0360i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-9.23 + 7.74i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.87 + 10.6i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.11 - 2.61i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.40 - 13.6i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.88 + 7.91i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.48 + 4.07i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.46 + 9.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.59 + 1.67i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (2.62 - 2.20i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.62783089061669474094402033937, −10.10444362124022206988649203935, −9.170829380475770159478112290327, −8.265886104531112597508371033430, −7.65647909106506252066261403518, −6.48037917170147806653173432159, −5.06693536038693487883148161659, −3.92011589457997167807281635921, −2.64199562196464181351459730107, −2.05318287754833519240498608088,
1.09397215213108229211984636618, 2.36490446701742030241754071108, 4.01134612039983924167963067799, 5.17003929856313793941520054728, 6.14237055800885408991183159807, 7.34879806017040543953637721796, 8.095977534778631161939099678076, 8.725385670276863346140930767166, 9.416071486786365512721566877315, 10.42458788240157456169708153811
