L(s) = 1 | + (−0.984 − 0.173i)2-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)5-s + (0.766 − 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (−0.866 + 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.984 − 0.173i)19-s + (0.342 + 0.939i)20-s + (0.642 − 0.766i)22-s + (0.866 − 0.5i)23-s + ⋯ |
L(s) = 1 | + (−0.984 − 0.173i)2-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)5-s + (0.766 − 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (−0.866 + 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.984 − 0.173i)19-s + (0.342 + 0.939i)20-s + (0.642 − 0.766i)22-s + (0.866 − 0.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7798372014 + 0.1433007768i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7798372014 + 0.1433007768i\) |
\(L(1)\) |
\(\approx\) |
\(0.8085682264 + 0.06292314655i\) |
\(L(1)\) |
\(\approx\) |
\(0.8085682264 + 0.06292314655i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 5 | \( 1 + (0.642 + 0.766i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.342 + 0.939i)T \) |
| 17 | \( 1 + (-0.342 - 0.939i)T \) |
| 19 | \( 1 + (0.984 - 0.173i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.766 - 0.642i)T \) |
| 59 | \( 1 + (-0.642 + 0.766i)T \) |
| 61 | \( 1 + (0.342 - 0.939i)T \) |
| 67 | \( 1 + (-0.766 + 0.642i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.642 - 0.766i)T \) |
| 83 | \( 1 + (0.939 - 0.342i)T \) |
| 89 | \( 1 + (0.642 - 0.766i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
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
−29.06631491683468333600946905946, −28.34090314479092905651255977224, −27.400569607693931125125482697954, −26.46292385515160346340758230135, −25.13947356024406709464067643357, −24.63891313540247987811649924634, −23.730310698051407918177393724406, −21.80877463129207535923830049216, −20.98633087947830010293584317362, −20.04513498462445303629344530877, −18.79377218753076268778750886014, −17.7879536694842373909049221279, −17.07610709642122220193464434191, −15.87047987808265170995197742336, −14.90556135858502907135755472407, −13.40951983719946088311248855404, −12.07329805247250960192036771645, −10.893977180664005191251102588479, −9.72363848637911072189805664913, −8.58389394533551996746499500222, −7.866438788124575028051487487999, −6.007652405766742567380460224915, −5.19282017914823457886283161235, −2.70880481319717501557452606604, −1.26028307963652550277133521912,
1.6464373788696772900984119986, 2.8944164257220388314972869801, 4.9126947965591876889641358361, 6.84134765537269193163907203242, 7.38626266110882323186309693794, 9.0038820458836038545992198250, 10.08170868597462998235344816377, 10.93656822997744357269613335810, 12.03822264480348451396514629388, 13.72160293189984965728297345219, 14.75848251387192823932257114471, 16.054821303754114952616352879157, 17.34037096346620856747819471796, 17.95500539479745191336192411815, 18.88135963141664319240356408235, 20.23934747794143055900203206122, 20.97557595982814824457854120326, 22.09348520909649813769464733661, 23.48764719904975708712890124656, 24.70857981520373786488905870838, 25.64352816676154203542722854785, 26.68390356807878780765536394311, 27.1375645491966111672194612028, 28.74212108496694315197923679087, 29.161457459560899106000631051975