L(s) = 1 | + (−0.623 + 0.781i)5-s + (0.222 − 0.974i)11-s + (0.733 + 0.680i)13-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.826 + 0.563i)29-s + (−0.5 − 0.866i)31-s + (0.826 + 0.563i)37-s + (−0.365 − 0.930i)41-s + (−0.365 + 0.930i)43-s + (−0.733 − 0.680i)47-s + (0.826 − 0.563i)53-s + (0.623 + 0.781i)55-s + ⋯ |
L(s) = 1 | + (−0.623 + 0.781i)5-s + (0.222 − 0.974i)11-s + (0.733 + 0.680i)13-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (0.826 + 0.563i)29-s + (−0.5 − 0.866i)31-s + (0.826 + 0.563i)37-s + (−0.365 − 0.930i)41-s + (−0.365 + 0.930i)43-s + (−0.733 − 0.680i)47-s + (0.826 − 0.563i)53-s + (0.623 + 0.781i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.324146764 + 0.4846223859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.324146764 + 0.4846223859i\) |
\(L(1)\) |
\(\approx\) |
\(1.007629706 + 0.1502205077i\) |
\(L(1)\) |
\(\approx\) |
\(1.007629706 + 0.1502205077i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.623 + 0.781i)T \) |
| 11 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.733 + 0.680i)T \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.900 + 0.433i)T \) |
| 29 | \( 1 + (0.826 + 0.563i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.826 + 0.563i)T \) |
| 41 | \( 1 + (-0.365 - 0.930i)T \) |
| 43 | \( 1 + (-0.365 + 0.930i)T \) |
| 47 | \( 1 + (-0.733 - 0.680i)T \) |
| 53 | \( 1 + (0.826 - 0.563i)T \) |
| 59 | \( 1 + (0.365 - 0.930i)T \) |
| 61 | \( 1 + (-0.0747 + 0.997i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (-0.955 - 0.294i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.733 + 0.680i)T \) |
| 89 | \( 1 + (0.733 - 0.680i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−20.13757282518010935184976038279, −19.64713463279258218916889495552, −18.61183484016363201957457159417, −18.0223275671822052694229952457, −17.11254652610206747703097387514, −16.464875957309739410579103012742, −15.72118117234293735755542579040, −15.11052027409372679056571072653, −14.289645136294926969259840429170, −13.27746498094998973874782225232, −12.64969961571065339898465796674, −12.0433872102734550208629660858, −11.22751044551854054644418310258, −10.36350171882966137134205567234, −9.49324055345538504887896743829, −8.71642993514136048661364392654, −8.01816169484091937114928837872, −7.23848266328979403469997265847, −6.34867680530688177390186587483, −5.2733211491857020548656611419, −4.62146029411434699039874072317, −3.8170381435288554688314423911, −2.853980579173338572346362817546, −1.65010932429435974847178455030, −0.69704312303510579093974934183,
0.869991545680840059816751434254, 2.09873601548001698666971226182, 3.167910117822786385179161010924, 3.76623604267461462208172234244, 4.638804589829837997592143110206, 5.85293864145914154058225889794, 6.55850215584538028770863453167, 7.174235935077217544998545163538, 8.3493148137640022629624698345, 8.68324130659169094699924786441, 9.811944302138947423893705002055, 10.828713494801419589896171024765, 11.208564371382581130067565913604, 11.86059314944209371533168141685, 13.04276052881876262632225738379, 13.56408194036574413601559211177, 14.570558171519159137661698641864, 15.03563655499592890369242770931, 15.90398641317493164422465270903, 16.55265079230046444817074731522, 17.3868986860862724634286768269, 18.26535887580310252907810844691, 18.98975215192763003496498051497, 19.39059619617037000381965868908, 20.15916377367236222830168354913