L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.917 − 2.03i)5-s + (0.499 + 0.866i)6-s − 3i·7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.81 − 1.30i)10-s + 2·11-s + 0.999i·12-s + (1.25 − 0.724i)13-s + (1.5 − 2.59i)14-s + (1.81 − 1.30i)15-s + (−0.5 + 0.866i)16-s + (−4.24 − 2.44i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.410 − 0.911i)5-s + (0.204 + 0.353i)6-s − 1.13i·7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.573 − 0.413i)10-s + 0.603·11-s + 0.288i·12-s + (0.348 − 0.201i)13-s + (0.400 − 0.694i)14-s + (0.468 − 0.337i)15-s + (−0.125 + 0.216i)16-s + (−1.02 − 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.60386 + 0.0278764i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.60386 + 0.0278764i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.917 + 2.03i)T \) |
| 19 | \( 1 + (-1 - 4.24i)T \) |
good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-1.25 + 0.724i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (4.24 + 2.44i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.12 + 1.22i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.67 - 4.63i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.89T + 31T^{2} \) |
| 37 | \( 1 - 9.44iT - 37T^{2} \) |
| 41 | \( 1 + (-1.22 + 2.12i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.71 + 2.72i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.31 - 4.22i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (11.1 - 6.44i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.89 - 6.75i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.17 - 8.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.98 + 1.72i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.22 + 7.31i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.81 - 1.05i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.39 - 2.42i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 13.3iT - 83T^{2} \) |
| 89 | \( 1 + (8.22 + 14.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.26 - 5.34i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.67465241776917985481734561884, −9.814927407029929389849672197864, −8.870829498150423814420353334796, −8.130582224155792673622607257482, −7.07703164652796298273487645355, −6.17638110854655654659117178146, −4.87834146842636039879014917158, −4.28231677495373254959296845418, −3.16515501129930089214983614345, −1.42250200333678320591460541703,
1.92281266071465925913366163613, 2.73389084361539852638982537517, 3.81582510883488983519931164169, 5.16738294005566397281733224218, 6.36114313885663746825373985340, 6.72655943323464913333047298746, 8.148788001909345191380758303772, 9.152370532152480570815722163500, 9.761897597985010322929677557302, 11.10061007818294894431854657654