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
−31.96414445747699878740643991268, −31.26668540234283272898094106867, −29.988032622282161394796727099028, −29.407965346188397342681745248613, −28.55815212893030206198620051738, −27.11471260931550599586725103970, −25.88922499004758635162486173653, −24.48284466109770582040578824796, −23.17781144889241122956950400227, −22.35267927481512998219862666481, −21.44002977497455218493114240620, −19.61131149458073349552832749922, −18.776068539536242607097527608896, −18.301442537749800825288023093326, −16.6002598520464949586929966357, −14.537268010710074631177375643738, −13.66561489312877153099070439154, −12.33045885568320788010402310950, −11.47760113910957694555745179406, −10.18485018624091653433701574904, −8.63930040107530066391018488142, −6.75313290058799439705315937561, −5.59353861156279606267522083743, −3.27275789783716310717692570797, −2.10507786765962107569849757401,
0.0864100535052616952583504307, 3.79607469271373463366713328838, 4.85772001901585284866501234682, 6.07870741998025174211209683035, 7.78495160162771906424914971694, 9.29451141823276885172174721340, 10.14133861103178060774186731335, 12.33155544411206586622769633265, 13.32862396496721478882767489349, 14.982616658111124075646471501324, 15.79892754466055410219646623233, 17.05754136636647042636252120821, 17.39013560706993942128828744848, 19.68786929967967567317623019130, 20.84686778227909130008971624244, 22.126907579538458001421168472840, 23.05875448558269953940399863102, 24.04344451055218423759592886073, 25.54286247414737399607875742386, 26.188214679592200637030534403982, 27.64701834129467563750376046876, 28.18622520843217082398543034047, 29.81063028716239459840217261023, 31.52556514363732635195812374689, 32.44253934146727130344936197468