L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s − i·11-s + (−0.5 − 0.866i)13-s + (−0.866 − 0.5i)15-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 + 0.5i)21-s + i·23-s + (−0.5 + 0.866i)25-s − i·27-s − 29-s − i·31-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s − i·11-s + (−0.5 − 0.866i)13-s + (−0.866 − 0.5i)15-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (0.866 + 0.5i)21-s + i·23-s + (−0.5 + 0.866i)25-s − i·27-s − 29-s − i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.04017505393 + 0.4098410829i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04017505393 + 0.4098410829i\) |
\(L(1)\) |
\(\approx\) |
\(0.6281839341 + 0.2240562852i\) |
\(L(1)\) |
\(\approx\) |
\(0.6281839341 + 0.2240562852i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + iT \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.866 + 0.5i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + iT \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.5594256472436402621589515914, −21.91596061660897552676828297044, −21.45672952343011882594758089601, −20.21334252445131119925580270113, −19.21537747226737265036081471354, −18.6226046509082766352104077327, −17.69260490695214439817861320989, −16.83104459213826899326589318728, −16.32300955444684058803636223180, −15.44622651016506782953300890426, −14.08351704199557390236172709906, −13.13982884208017164288568637037, −12.68907758573689762778787726579, −11.6787797804892840731786491173, −11.05525445969978774362766195019, −9.70293931145670824659951707764, −9.0030792674717560873695866786, −8.06378211663008860961318008508, −6.67560686574617863994382444858, −6.07750231438391350214424370627, −5.220677061855954881913811141489, −4.33552446769515454565557864890, −2.619470488851712421906502706949, −1.66840645451741902949809046593, −0.22878554356141947538921956163,
1.56151420471413117183009658454, 3.03954781409166457433887989305, 3.95916422843286623200359708347, 5.08628608451059759256687991615, 5.954659959186352021122442444431, 7.02059940715729781339242319136, 7.44105554975323316182573201929, 9.35408441398957551543311659816, 9.99642522051389685362585476671, 10.54861474461944098716990779124, 11.43768247011158853955044265443, 12.48452402675133418082054890771, 13.3325328679703178106684854904, 14.37098132013020517323824584239, 15.24042334288361856207720611788, 15.975560565219916669641309794557, 17.03171229636085648325621941949, 17.67697949265261613940929281253, 18.18016140137226672510410506116, 19.4353517617735521054204613511, 20.34696372848686645302693160417, 21.08294489565458700846513977541, 22.29762238065587107408349130504, 22.55018287776002264299580776861, 23.15933245622598562547559569729