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 | \( 1 + (0.190 + 0.981i)T \) |
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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
−31.01499119356857955299324003106, −30.32596477958914174997634200691, −28.94870123287652824485674636766, −27.92556048743298247804539581506, −26.43350533052393000321295354348, −26.084721715258716841797297221728, −24.23456832864036386869331664207, −23.108250077917351051955271343145, −22.599427271593828626169547213831, −21.306490305998828502424411268350, −20.48817245798705809916531996082, −19.5423524813231090520027150669, −17.831741893037537287475431308747, −16.1229718767728365026191440831, −15.56818561812585859507644357247, −14.338928013808495299689105407775, −13.57332760651572414932579921429, −11.6201126265342782763873230237, −10.94269341381814825082821324500, −9.93429798300219437969562612003, −7.806380734519246146777197841855, −6.54063113680052725115699522046, −4.807662098246490972963196481307, −3.95425244162488348585409335152, −2.6550867649065598736766105924,
1.80443109893056413378980024669, 3.38742415724186439137773878052, 5.234290068822049768744203035, 6.06997908512940945938670466691, 7.85442576254966617491327890326, 8.4491633877780824062597268080, 11.03873717133513308462856324693, 12.12974177657787642140341701147, 12.836156570257773594322270540131, 13.85516060949972597800755034257, 15.30169193709993051623217769849, 16.16002044427349629991231694899, 17.65545760466743444488910111357, 18.91132897722617546963798149416, 20.138261121018011261088881692484, 21.00122006836674972040037177318, 22.38127156819892527284869312715, 23.54414925017461772419075773759, 24.24976789086925166564549953114, 24.944009924188600804350173726215, 26.07505337034195494755322004249, 28.0162496745319578590580106467, 28.80421300746352699700910698370, 29.91391273502495043625123777489, 31.19340969297100560320933123460