L(s) = 1 | + (−0.477 − 0.878i)2-s + (−0.264 + 0.964i)3-s + (−0.543 + 0.839i)4-s + (0.771 + 0.636i)5-s + (0.973 − 0.227i)6-s + (−0.665 + 0.746i)7-s + (0.997 + 0.0765i)8-s + (−0.859 − 0.511i)9-s + (0.190 − 0.981i)10-s + (0.817 + 0.575i)11-s + (−0.665 − 0.746i)12-s + (−0.606 − 0.795i)13-s + (0.973 + 0.227i)14-s + (−0.817 + 0.575i)15-s + (−0.409 − 0.912i)16-s + (−0.927 − 0.373i)17-s + ⋯ |
L(s) = 1 | + (−0.477 − 0.878i)2-s + (−0.264 + 0.964i)3-s + (−0.543 + 0.839i)4-s + (0.771 + 0.636i)5-s + (0.973 − 0.227i)6-s + (−0.665 + 0.746i)7-s + (0.997 + 0.0765i)8-s + (−0.859 − 0.511i)9-s + (0.190 − 0.981i)10-s + (0.817 + 0.575i)11-s + (−0.665 − 0.746i)12-s + (−0.606 − 0.795i)13-s + (0.973 + 0.227i)14-s + (−0.817 + 0.575i)15-s + (−0.409 − 0.912i)16-s + (−0.927 − 0.373i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.840 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1578138142 + 0.5355270209i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1578138142 + 0.5355270209i\) |
\(L(1)\) |
\(\approx\) |
\(0.6240747891 + 0.1693289321i\) |
\(L(1)\) |
\(\approx\) |
\(0.6240747891 + 0.1693289321i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 83 | \( 1 \) |
good | 2 | \( 1 + (-0.477 - 0.878i)T \) |
| 3 | \( 1 + (-0.264 + 0.964i)T \) |
| 5 | \( 1 + (0.771 + 0.636i)T \) |
| 7 | \( 1 + (-0.665 + 0.746i)T \) |
| 11 | \( 1 + (0.817 + 0.575i)T \) |
| 13 | \( 1 + (-0.606 - 0.795i)T \) |
| 17 | \( 1 + (-0.927 - 0.373i)T \) |
| 19 | \( 1 + (-0.988 - 0.152i)T \) |
| 23 | \( 1 + (0.0383 + 0.999i)T \) |
| 29 | \( 1 + (0.953 + 0.301i)T \) |
| 31 | \( 1 + (-0.997 + 0.0765i)T \) |
| 37 | \( 1 + (-0.859 + 0.511i)T \) |
| 41 | \( 1 + (0.477 - 0.878i)T \) |
| 43 | \( 1 + (-0.720 - 0.693i)T \) |
| 47 | \( 1 + (-0.896 + 0.443i)T \) |
| 53 | \( 1 + (-0.896 - 0.443i)T \) |
| 59 | \( 1 + (0.338 - 0.941i)T \) |
| 61 | \( 1 + (-0.114 + 0.993i)T \) |
| 67 | \( 1 + (0.409 + 0.912i)T \) |
| 71 | \( 1 + (0.665 + 0.746i)T \) |
| 73 | \( 1 + (-0.190 + 0.981i)T \) |
| 79 | \( 1 + (0.543 - 0.839i)T \) |
| 89 | \( 1 + (0.973 - 0.227i)T \) |
| 97 | \( 1 + (0.973 + 0.227i)T \) |
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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
−29.82904836149964603153154964003, −29.055400786793409522889664622027, −28.22510064513022730376694811519, −26.752736455862280443891546258409, −25.74588210652961064095582866192, −24.72709360851899738472609053289, −24.12381025362066845598413994786, −23.03705040214890759089286884834, −21.86324094574612924659048812265, −19.881397911162342926040234657421, −19.231069476992137989554146135837, −17.886356370651535641333706583933, −16.8996312175034119899318067736, −16.5010784454802304289697344233, −14.44295043847983407727638925402, −13.59816383157665702594474723391, −12.611575553517269178975474398047, −10.78727462263045940658793164592, −9.36226316821814722196856419716, −8.34265128499479142507599545329, −6.70866517148114511516563318477, −6.266113702469928594948887661678, −4.58171141734930812407452394290, −1.77176701336330068817956063502, −0.30946779166315423932375272526,
2.29090703504199231744770241388, 3.49196417235742540678072091702, 5.12205335202899631467497598481, 6.71023958046720511213955444918, 8.87300841345239453680426584155, 9.669107727842619876009178313982, 10.53791764427176780970580380001, 11.73825685659546285538639090493, 12.96030061515119695206988232145, 14.50840552199998896972430632667, 15.69822068837188322763937830291, 17.23736079179610280229798541731, 17.785659445910432776548427178218, 19.25514188594825257837702767213, 20.25154271509993094769880411415, 21.569609688358520623954076607716, 22.1308138049885438637321721868, 22.82330400097772651634069967776, 25.288999530662053533619472764877, 25.83468424152616588931794588955, 27.10878117893540894043245989267, 27.83763043276057856100314084967, 28.97237155018246077485718437257, 29.61161851912314249722064915097, 30.95886780002887026584766468864