L(s) = 1 | + (0.207 − 0.978i)2-s + (−0.309 − 0.951i)3-s + (−0.913 − 0.406i)4-s + (0.104 + 0.994i)5-s + (−0.994 + 0.104i)6-s + (−0.743 + 0.669i)7-s + (−0.587 + 0.809i)8-s + (−0.809 + 0.587i)9-s + (0.994 + 0.104i)10-s − i·11-s + (−0.104 + 0.994i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.913 − 0.406i)15-s + (0.669 + 0.743i)16-s + (0.406 − 0.913i)17-s + ⋯ |
L(s) = 1 | + (0.207 − 0.978i)2-s + (−0.309 − 0.951i)3-s + (−0.913 − 0.406i)4-s + (0.104 + 0.994i)5-s + (−0.994 + 0.104i)6-s + (−0.743 + 0.669i)7-s + (−0.587 + 0.809i)8-s + (−0.809 + 0.587i)9-s + (0.994 + 0.104i)10-s − i·11-s + (−0.104 + 0.994i)12-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)14-s + (0.913 − 0.406i)15-s + (0.669 + 0.743i)16-s + (0.406 − 0.913i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.183 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.183 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1336185273 + 0.1109673565i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1336185273 + 0.1109673565i\) |
\(L(1)\) |
\(\approx\) |
\(0.5701189762 - 0.3166997891i\) |
\(L(1)\) |
\(\approx\) |
\(0.5701189762 - 0.3166997891i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 \) |
good | 2 | \( 1 + (0.207 - 0.978i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.104 + 0.994i)T \) |
| 7 | \( 1 + (-0.743 + 0.669i)T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.406 - 0.913i)T \) |
| 19 | \( 1 + (-0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.587 - 0.809i)T \) |
| 29 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (0.207 + 0.978i)T \) |
| 37 | \( 1 + (-0.951 - 0.309i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + (-0.406 - 0.913i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.587 - 0.809i)T \) |
| 59 | \( 1 + (-0.207 + 0.978i)T \) |
| 67 | \( 1 + (-0.994 + 0.104i)T \) |
| 71 | \( 1 + (0.994 + 0.104i)T \) |
| 73 | \( 1 + (-0.104 + 0.994i)T \) |
| 79 | \( 1 + (0.406 + 0.913i)T \) |
| 83 | \( 1 + (-0.978 - 0.207i)T \) |
| 89 | \( 1 + (0.951 - 0.309i)T \) |
| 97 | \( 1 + (0.978 - 0.207i)T \) |
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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
−32.44253934146727130344936197468, −31.52556514363732635195812374689, −29.81063028716239459840217261023, −28.18622520843217082398543034047, −27.64701834129467563750376046876, −26.188214679592200637030534403982, −25.54286247414737399607875742386, −24.04344451055218423759592886073, −23.05875448558269953940399863102, −22.126907579538458001421168472840, −20.84686778227909130008971624244, −19.68786929967967567317623019130, −17.39013560706993942128828744848, −17.05754136636647042636252120821, −15.79892754466055410219646623233, −14.982616658111124075646471501324, −13.32862396496721478882767489349, −12.33155544411206586622769633265, −10.14133861103178060774186731335, −9.29451141823276885172174721340, −7.78495160162771906424914971694, −6.07870741998025174211209683035, −4.85772001901585284866501234682, −3.79607469271373463366713328838, −0.0864100535052616952583504307,
2.10507786765962107569849757401, 3.27275789783716310717692570797, 5.59353861156279606267522083743, 6.75313290058799439705315937561, 8.63930040107530066391018488142, 10.18485018624091653433701574904, 11.47760113910957694555745179406, 12.33045885568320788010402310950, 13.66561489312877153099070439154, 14.537268010710074631177375643738, 16.6002598520464949586929966357, 18.301442537749800825288023093326, 18.776068539536242607097527608896, 19.61131149458073349552832749922, 21.44002977497455218493114240620, 22.35267927481512998219862666481, 23.17781144889241122956950400227, 24.48284466109770582040578824796, 25.88922499004758635162486173653, 27.11471260931550599586725103970, 28.55815212893030206198620051738, 29.407965346188397342681745248613, 29.988032622282161394796727099028, 31.26668540234283272898094106867, 31.96414445747699878740643991268