L(s) = 1 | + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.866 + 0.5i)8-s + (−0.766 + 0.642i)11-s + (−0.342 − 0.939i)13-s + (−0.939 − 0.342i)16-s + i·17-s − 19-s + (−0.984 − 0.173i)22-s + (0.342 + 0.939i)23-s + (0.5 − 0.866i)26-s + (−0.939 − 0.342i)29-s + (0.173 − 0.984i)31-s + (−0.342 − 0.939i)32-s + (−0.766 + 0.642i)34-s + ⋯ |
L(s) = 1 | + (0.642 + 0.766i)2-s + (−0.173 + 0.984i)4-s + (−0.866 + 0.5i)8-s + (−0.766 + 0.642i)11-s + (−0.342 − 0.939i)13-s + (−0.939 − 0.342i)16-s + i·17-s − 19-s + (−0.984 − 0.173i)22-s + (0.342 + 0.939i)23-s + (0.5 − 0.866i)26-s + (−0.939 − 0.342i)29-s + (0.173 − 0.984i)31-s + (−0.342 − 0.939i)32-s + (−0.766 + 0.642i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2188682689 + 0.5804730216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2188682689 + 0.5804730216i\) |
\(L(1)\) |
\(\approx\) |
\(0.8250409007 + 0.5951124928i\) |
\(L(1)\) |
\(\approx\) |
\(0.8250409007 + 0.5951124928i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.642 + 0.766i)T \) |
| 11 | \( 1 + (-0.766 + 0.642i)T \) |
| 13 | \( 1 + (-0.342 - 0.939i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.342 + 0.939i)T \) |
| 29 | \( 1 + (-0.939 - 0.342i)T \) |
| 31 | \( 1 + (0.173 - 0.984i)T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.984 + 0.173i)T \) |
| 47 | \( 1 + (-0.984 + 0.173i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (0.173 + 0.984i)T \) |
| 67 | \( 1 + (0.642 - 0.766i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.766 + 0.642i)T \) |
| 83 | \( 1 + (-0.342 + 0.939i)T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + (0.984 - 0.173i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.36670291394997016975464537074, −20.68644446566928657627415211900, −19.88078839214504530544190984004, −18.86424651898937669098581820620, −18.658570783479123185121567097921, −17.50782975782521979304819883228, −16.37755184661632364101151757048, −15.75515783232199376549394317029, −14.596249595523227676152717305111, −14.177355617075137402392132189560, −13.17059528280571028049562738538, −12.60102514922954927592268381516, −11.59973174785649685474148688664, −10.973394399199109062614385631437, −10.17375288139988114292241846345, −9.229193925549366534151849938, −8.45436073720572740528471745246, −7.08641597699796022728972845656, −6.29961378307153837506118396811, −5.19151016430971262242610643753, −4.59800054134929327282526993142, −3.47294954523635362974220892405, −2.60706390251831374224168744594, −1.69130028231572113833889989152, −0.19540177750356852217952709318,
1.93972741229361123282763883697, 2.99577575517977817649865282721, 3.96900216481051842485320140829, 4.901847418002756742914278101201, 5.68510241993858432154247495813, 6.514253689144637767282920785542, 7.6375656430157712453093150512, 8.00973776723240511064435779022, 9.129138284905627645156231177027, 10.15629566268014645501919096079, 11.07908554062054837398746919903, 12.17135348826163110155005342488, 12.96207756545754754993555918427, 13.31874195292794877255007098514, 14.581691657555587657792687628128, 15.2154795613810026989419767385, 15.61374213908023673168251631103, 16.87913047658837652044562902434, 17.31521374904152244999136455536, 18.10816370502396483453107487966, 19.105764198792933089496781964853, 20.1106122984328190355005223313, 20.97551476777675590498327027464, 21.53059027069220227198367927878, 22.537585411826573105829374684001