L(s) = 1 | + (0.988 − 0.152i)2-s + (0.0383 + 0.999i)3-s + (0.953 − 0.301i)4-s + (−0.543 + 0.839i)5-s + (0.190 + 0.981i)6-s + (0.338 − 0.941i)7-s + (0.896 − 0.443i)8-s + (−0.997 + 0.0765i)9-s + (−0.409 + 0.912i)10-s + (−0.859 + 0.511i)11-s + (0.338 + 0.941i)12-s + (0.720 + 0.693i)13-s + (0.190 − 0.981i)14-s + (−0.859 − 0.511i)15-s + (0.817 − 0.575i)16-s + (−0.665 − 0.746i)17-s + ⋯ |
L(s) = 1 | + (0.988 − 0.152i)2-s + (0.0383 + 0.999i)3-s + (0.953 − 0.301i)4-s + (−0.543 + 0.839i)5-s + (0.190 + 0.981i)6-s + (0.338 − 0.941i)7-s + (0.896 − 0.443i)8-s + (−0.997 + 0.0765i)9-s + (−0.409 + 0.912i)10-s + (−0.859 + 0.511i)11-s + (0.338 + 0.941i)12-s + (0.720 + 0.693i)13-s + (0.190 − 0.981i)14-s + (−0.859 − 0.511i)15-s + (0.817 − 0.575i)16-s + (−0.665 − 0.746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.479267430 + 0.5474772023i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.479267430 + 0.5474772023i\) |
\(L(1)\) |
\(\approx\) |
\(1.560063876 + 0.3738821907i\) |
\(L(1)\) |
\(\approx\) |
\(1.560063876 + 0.3738821907i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (0.988 - 0.152i)T \) |
| 3 | \( 1 + (0.0383 + 0.999i)T \) |
| 5 | \( 1 + (-0.543 + 0.839i)T \) |
| 7 | \( 1 + (0.338 - 0.941i)T \) |
| 11 | \( 1 + (-0.859 + 0.511i)T \) |
| 13 | \( 1 + (0.720 + 0.693i)T \) |
| 17 | \( 1 + (-0.665 - 0.746i)T \) |
| 19 | \( 1 + (0.606 - 0.795i)T \) |
| 23 | \( 1 + (-0.973 - 0.227i)T \) |
| 29 | \( 1 + (-0.264 - 0.964i)T \) |
| 31 | \( 1 + (0.896 + 0.443i)T \) |
| 37 | \( 1 + (-0.997 - 0.0765i)T \) |
| 41 | \( 1 + (0.988 + 0.152i)T \) |
| 43 | \( 1 + (-0.114 + 0.993i)T \) |
| 47 | \( 1 + (-0.927 + 0.373i)T \) |
| 53 | \( 1 + (-0.927 - 0.373i)T \) |
| 59 | \( 1 + (0.477 + 0.878i)T \) |
| 61 | \( 1 + (-0.771 + 0.636i)T \) |
| 67 | \( 1 + (0.817 - 0.575i)T \) |
| 71 | \( 1 + (0.338 + 0.941i)T \) |
| 73 | \( 1 + (-0.409 + 0.912i)T \) |
| 79 | \( 1 + (0.953 - 0.301i)T \) |
| 89 | \( 1 + (0.190 + 0.981i)T \) |
| 97 | \( 1 + (0.190 - 0.981i)T \) |
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
−31.19340969297100560320933123460, −29.91391273502495043625123777489, −28.80421300746352699700910698370, −28.0162496745319578590580106467, −26.07505337034195494755322004249, −24.944009924188600804350173726215, −24.24976789086925166564549953114, −23.54414925017461772419075773759, −22.38127156819892527284869312715, −21.00122006836674972040037177318, −20.138261121018011261088881692484, −18.91132897722617546963798149416, −17.65545760466743444488910111357, −16.16002044427349629991231694899, −15.30169193709993051623217769849, −13.85516060949972597800755034257, −12.836156570257773594322270540131, −12.12974177657787642140341701147, −11.03873717133513308462856324693, −8.4491633877780824062597268080, −7.85442576254966617491327890326, −6.06997908512940945938670466691, −5.234290068822049768744203035, −3.38742415724186439137773878052, −1.80443109893056413378980024669,
2.6550867649065598736766105924, 3.95425244162488348585409335152, 4.807662098246490972963196481307, 6.54063113680052725115699522046, 7.806380734519246146777197841855, 9.93429798300219437969562612003, 10.94269341381814825082821324500, 11.6201126265342782763873230237, 13.57332760651572414932579921429, 14.338928013808495299689105407775, 15.56818561812585859507644357247, 16.1229718767728365026191440831, 17.831741893037537287475431308747, 19.5423524813231090520027150669, 20.48817245798705809916531996082, 21.306490305998828502424411268350, 22.599427271593828626169547213831, 23.108250077917351051955271343145, 24.23456832864036386869331664207, 26.084721715258716841797297221728, 26.43350533052393000321295354348, 27.92556048743298247804539581506, 28.94870123287652824485674636766, 30.32596477958914174997634200691, 31.01499119356857955299324003106