L(s) = 1 | + (0.859 − 0.511i)2-s + (0.606 − 0.795i)3-s + (0.477 − 0.878i)4-s + (−0.338 − 0.941i)5-s + (0.114 − 0.993i)6-s + (−0.409 + 0.912i)7-s + (−0.0383 − 0.999i)8-s + (−0.264 − 0.964i)9-s + (−0.771 − 0.636i)10-s + (0.953 − 0.301i)11-s + (−0.409 − 0.912i)12-s + (−0.896 + 0.443i)13-s + (0.114 + 0.993i)14-s + (−0.953 − 0.301i)15-s + (−0.543 − 0.839i)16-s + (0.190 + 0.981i)17-s + ⋯ |
L(s) = 1 | + (0.859 − 0.511i)2-s + (0.606 − 0.795i)3-s + (0.477 − 0.878i)4-s + (−0.338 − 0.941i)5-s + (0.114 − 0.993i)6-s + (−0.409 + 0.912i)7-s + (−0.0383 − 0.999i)8-s + (−0.264 − 0.964i)9-s + (−0.771 − 0.636i)10-s + (0.953 − 0.301i)11-s + (−0.409 − 0.912i)12-s + (−0.896 + 0.443i)13-s + (0.114 + 0.993i)14-s + (−0.953 − 0.301i)15-s + (−0.543 − 0.839i)16-s + (0.190 + 0.981i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.190627826 - 2.701703913i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.190627826 - 2.701703913i\) |
\(L(1)\) |
\(\approx\) |
\(1.422416022 - 1.303980132i\) |
\(L(1)\) |
\(\approx\) |
\(1.422416022 - 1.303980132i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (0.859 - 0.511i)T \) |
| 3 | \( 1 + (0.606 - 0.795i)T \) |
| 5 | \( 1 + (-0.338 - 0.941i)T \) |
| 7 | \( 1 + (-0.409 + 0.912i)T \) |
| 11 | \( 1 + (0.953 - 0.301i)T \) |
| 13 | \( 1 + (-0.896 + 0.443i)T \) |
| 17 | \( 1 + (0.190 + 0.981i)T \) |
| 19 | \( 1 + (0.997 - 0.0765i)T \) |
| 23 | \( 1 + (0.720 - 0.693i)T \) |
| 29 | \( 1 + (0.988 - 0.152i)T \) |
| 31 | \( 1 + (0.0383 - 0.999i)T \) |
| 37 | \( 1 + (-0.264 + 0.964i)T \) |
| 41 | \( 1 + (-0.859 - 0.511i)T \) |
| 43 | \( 1 + (0.927 - 0.373i)T \) |
| 47 | \( 1 + (0.973 + 0.227i)T \) |
| 53 | \( 1 + (0.973 - 0.227i)T \) |
| 59 | \( 1 + (0.817 + 0.575i)T \) |
| 61 | \( 1 + (-0.665 + 0.746i)T \) |
| 67 | \( 1 + (0.543 + 0.839i)T \) |
| 71 | \( 1 + (0.409 + 0.912i)T \) |
| 73 | \( 1 + (0.771 + 0.636i)T \) |
| 79 | \( 1 + (-0.477 + 0.878i)T \) |
| 89 | \( 1 + (0.114 - 0.993i)T \) |
| 97 | \( 1 + (0.114 + 0.993i)T \) |
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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
−31.12850142545491899925853337208, −30.26451939226093076555833775243, −29.32251264491744650642689168389, −27.11315156957506612086512735673, −26.832704746768379788198915957693, −25.60240340558927067260969702690, −24.811607489912183894553152189803, −23.11124448633238648341257765705, −22.54441109065849329102358007232, −21.60967499889430383179827639473, −20.21656128450047312283319429860, −19.55950199579449735898795521509, −17.55331277468028003191039937488, −16.36271396364798942806310240498, −15.40298218164980078413829180246, −14.35220032059905132108601847584, −13.78692744594200478091995588376, −12.03773099841398710120824040013, −10.74754788991229773617634721761, −9.4876623286758008185243253067, −7.646258795327857433042390277104, −6.89771552828949544768061435618, −5.018160580152560026489749472913, −3.7294935965346957615149895356, −2.88637607302770311501198417840,
1.086249520778363380960601608811, 2.5507050098513907504744005248, 3.98871909635283908697027044948, 5.58647515334328137765028673573, 6.87025375928984258137322535572, 8.59449482649473331382325034444, 9.60530380827554693316163153644, 11.83415472830508323761921570241, 12.2501163180642180049638414412, 13.3228035070395106831249532899, 14.50067333954766343517305377531, 15.51061749759665674027097319618, 16.98956456413325461120846594776, 18.85467948607444152381093197844, 19.46034135929010792445497243088, 20.385881570901531773936924510481, 21.561973905244997024535445844267, 22.672208310863933486087019489866, 24.13798370883986925107063924592, 24.4800709256414030995755736193, 25.49381351695260610797132280096, 27.26349598483634921371422392946, 28.6118893909640142219867038056, 29.15111472844542517647512607292, 30.51594601873476134229251194483