L(s) = 1 | + (0.962 − 0.269i)2-s + (0.962 − 0.269i)3-s + (0.854 − 0.519i)4-s + (−0.917 + 0.398i)5-s + (0.854 − 0.519i)6-s + (0.854 + 0.519i)7-s + (0.682 − 0.730i)8-s + (0.854 − 0.519i)9-s + (−0.775 + 0.631i)10-s + (0.203 + 0.979i)11-s + (0.682 − 0.730i)12-s + (−0.775 − 0.631i)13-s + (0.962 + 0.269i)14-s + (−0.775 + 0.631i)15-s + (0.460 − 0.887i)16-s + (−0.775 − 0.631i)17-s + ⋯ |
L(s) = 1 | + (0.962 − 0.269i)2-s + (0.962 − 0.269i)3-s + (0.854 − 0.519i)4-s + (−0.917 + 0.398i)5-s + (0.854 − 0.519i)6-s + (0.854 + 0.519i)7-s + (0.682 − 0.730i)8-s + (0.854 − 0.519i)9-s + (−0.775 + 0.631i)10-s + (0.203 + 0.979i)11-s + (0.682 − 0.730i)12-s + (−0.775 − 0.631i)13-s + (0.962 + 0.269i)14-s + (−0.775 + 0.631i)15-s + (0.460 − 0.887i)16-s + (−0.775 − 0.631i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.205685697 - 0.8165896653i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.205685697 - 0.8165896653i\) |
\(L(1)\) |
\(\approx\) |
\(2.312312158 - 0.4344563895i\) |
\(L(1)\) |
\(\approx\) |
\(2.312312158 - 0.4344563895i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.962 - 0.269i)T \) |
| 3 | \( 1 + (0.962 - 0.269i)T \) |
| 5 | \( 1 + (-0.917 + 0.398i)T \) |
| 7 | \( 1 + (0.854 + 0.519i)T \) |
| 11 | \( 1 + (0.203 + 0.979i)T \) |
| 13 | \( 1 + (-0.775 - 0.631i)T \) |
| 17 | \( 1 + (-0.775 - 0.631i)T \) |
| 19 | \( 1 + (0.962 - 0.269i)T \) |
| 29 | \( 1 + (0.203 + 0.979i)T \) |
| 31 | \( 1 + (-0.576 + 0.816i)T \) |
| 37 | \( 1 + (0.460 + 0.887i)T \) |
| 41 | \( 1 + (-0.334 + 0.942i)T \) |
| 43 | \( 1 + (-0.0682 - 0.997i)T \) |
| 47 | \( 1 + (-0.576 - 0.816i)T \) |
| 53 | \( 1 + (-0.775 - 0.631i)T \) |
| 59 | \( 1 + (0.962 + 0.269i)T \) |
| 61 | \( 1 + (-0.917 + 0.398i)T \) |
| 67 | \( 1 + (0.203 + 0.979i)T \) |
| 71 | \( 1 + (-0.334 - 0.942i)T \) |
| 73 | \( 1 + (0.203 - 0.979i)T \) |
| 79 | \( 1 + (-0.0682 - 0.997i)T \) |
| 83 | \( 1 + (-0.917 + 0.398i)T \) |
| 89 | \( 1 + (-0.990 - 0.136i)T \) |
| 97 | \( 1 + (-0.334 - 0.942i)T \) |
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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
−23.87104194950127393440532494879, −22.70472199526372681989213199051, −21.74218335409519004364238591717, −21.108357932478382922123706928178, −20.237296743043541910833983587464, −19.72467016544490722672015278933, −18.85652247777317650708761866386, −17.289859879960804305245659280114, −16.440375781290035404119286540199, −15.741602789222134609697379345297, −14.85713798708235232940685812278, −14.2265525927424357345658368050, −13.50455181437983795290672818599, −12.53562279767309532965594063353, −11.4727084999523402529736850515, −10.916763545652152269731356191621, −9.412247595992014136937259405004, −8.24280463224982112521992777972, −7.80119852624542958944686571105, −6.878529115563087057402936342564, −5.394022936335828976391337773414, −4.28988147172846234840182397601, −3.97523789269519345262727625435, −2.78154264202211578200415776396, −1.54945324817881846286663482547,
1.49330705405257087343013675190, 2.57121699957725734424026528995, 3.275232629088289901086628303110, 4.49219206883336513578350483036, 5.1007145040430908339069985278, 6.863912555013399333262407344405, 7.32119560711809956866785983641, 8.28317450189534390351614140044, 9.500578405100377941772181699965, 10.55273325896680080463749794647, 11.69641824768897169751038025836, 12.1802058909406825750140569461, 13.12172923540446113415599350818, 14.18690701624146741344206835308, 14.88028495094182585385827483691, 15.256602710999176022708256842761, 16.13537881647713266307320174498, 17.86553985541654534647635118806, 18.4782030053619587564509549384, 19.69320426036837704703693446849, 20.05813962652660226389591055339, 20.701390475025530588032258490561, 21.91174224334812732543978914875, 22.413999358278820787424191039529, 23.54643422628939857192540051531