L(s) = 1 | + (0.753 + 0.657i)2-s + (0.473 + 0.880i)3-s + (0.134 + 0.990i)4-s + (0.134 − 0.990i)5-s + (−0.222 + 0.974i)6-s + (0.473 − 0.880i)7-s + (−0.550 + 0.834i)8-s + (−0.550 + 0.834i)9-s + (0.753 − 0.657i)10-s + (0.983 − 0.178i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (0.936 − 0.351i)14-s + (0.936 − 0.351i)15-s + (−0.963 + 0.266i)16-s + (−0.393 − 0.919i)17-s + ⋯ |
L(s) = 1 | + (0.753 + 0.657i)2-s + (0.473 + 0.880i)3-s + (0.134 + 0.990i)4-s + (0.134 − 0.990i)5-s + (−0.222 + 0.974i)6-s + (0.473 − 0.880i)7-s + (−0.550 + 0.834i)8-s + (−0.550 + 0.834i)9-s + (0.753 − 0.657i)10-s + (0.983 − 0.178i)11-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (0.936 − 0.351i)14-s + (0.936 − 0.351i)15-s + (−0.963 + 0.266i)16-s + (−0.393 − 0.919i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0527 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 421 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0527 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.829613244 + 1.735549956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.829613244 + 1.735549956i\) |
\(L(1)\) |
\(\approx\) |
\(1.637785373 + 0.9813806805i\) |
\(L(1)\) |
\(\approx\) |
\(1.637785373 + 0.9813806805i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 421 | \( 1 \) |
good | 2 | \( 1 + (0.753 + 0.657i)T \) |
| 3 | \( 1 + (0.473 + 0.880i)T \) |
| 5 | \( 1 + (0.134 - 0.990i)T \) |
| 7 | \( 1 + (0.473 - 0.880i)T \) |
| 11 | \( 1 + (0.983 - 0.178i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.393 - 0.919i)T \) |
| 19 | \( 1 + (0.858 + 0.512i)T \) |
| 23 | \( 1 + (-0.691 + 0.722i)T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + (-0.0448 + 0.998i)T \) |
| 37 | \( 1 + (0.983 - 0.178i)T \) |
| 41 | \( 1 + (0.858 + 0.512i)T \) |
| 43 | \( 1 + (-0.963 + 0.266i)T \) |
| 47 | \( 1 + (0.134 - 0.990i)T \) |
| 53 | \( 1 + (0.473 - 0.880i)T \) |
| 59 | \( 1 + (-0.995 - 0.0896i)T \) |
| 61 | \( 1 + (-0.963 + 0.266i)T \) |
| 67 | \( 1 + (0.309 - 0.951i)T \) |
| 71 | \( 1 + (-0.995 + 0.0896i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.963 - 0.266i)T \) |
| 83 | \( 1 + (-0.995 - 0.0896i)T \) |
| 89 | \( 1 + (-0.0448 - 0.998i)T \) |
| 97 | \( 1 + (0.858 - 0.512i)T \) |
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
−24.02468469987803070659094243381, −22.98067734512712071830678386864, −22.24209546163113198014198463916, −21.6237663531126122402904237261, −20.39683516351104719836508294144, −19.76823750817783803701906154523, −18.88823378266813374187544114292, −18.182192366404461835346248845778, −17.54856790923958360218206924131, −15.55179962996355872177809530933, −14.896808119792535321620536069334, −14.273087557094733974830411591467, −13.433548685793294120190871240378, −12.43618946767773385573083515476, −11.72861111202194244193475985321, −10.902008221358939769931948839598, −9.73299327716299441613682186265, −8.68691913492417009139104381223, −7.51362747783250576718088188409, −6.279247340594820663649579725620, −5.89489025397688351991420387536, −4.24410546427849548698680649089, −3.05225983852889033310897057694, −2.38210096077475340810193910375, −1.31027049867655754900480530612,
1.61914423784042823010538865962, 3.29548299542690821603833920166, 4.26860043588013024097562998498, 4.71511382671953441182251913536, 5.85151876795512401347306453812, 7.13059989746272443802041580242, 8.17004193734392883509941642630, 8.97177747633002692395620205213, 9.82238864104166058713906134321, 11.36543697716954753469327279264, 11.925582235431205908548962789879, 13.449215625029460856924842735217, 13.93893633585676536166302742543, 14.54723101865726238143554848088, 15.907033646699530178637961318935, 16.33402235269732767644997616676, 17.00821572539599689128408537856, 17.9729468078423304905064620644, 19.96730893369142307101183392613, 20.07176917652141186576875458524, 21.3093554347871565565188366611, 21.55982749065152303868392468217, 22.79893545025705372496882117600, 23.54042479969757331263689541942, 24.59292469545385657898203289569