L(s) = 1 | + (−0.816 − 0.577i)2-s + (0.332 + 0.943i)4-s + (−0.552 + 0.833i)5-s + (0.969 + 0.243i)7-s + (0.273 − 0.961i)8-s + (0.932 − 0.361i)10-s + (−0.816 − 0.577i)11-s + (−0.696 + 0.717i)13-s + (−0.650 − 0.759i)14-s + (−0.779 + 0.626i)16-s + (0.602 − 0.798i)17-s + (0.739 + 0.673i)19-s + (−0.969 − 0.243i)20-s + (0.332 + 0.943i)22-s + (−0.816 + 0.577i)23-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.577i)2-s + (0.332 + 0.943i)4-s + (−0.552 + 0.833i)5-s + (0.969 + 0.243i)7-s + (0.273 − 0.961i)8-s + (0.932 − 0.361i)10-s + (−0.816 − 0.577i)11-s + (−0.696 + 0.717i)13-s + (−0.650 − 0.759i)14-s + (−0.779 + 0.626i)16-s + (0.602 − 0.798i)17-s + (0.739 + 0.673i)19-s + (−0.969 − 0.243i)20-s + (0.332 + 0.943i)22-s + (−0.816 + 0.577i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.132486703 - 0.1146623839i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132486703 - 0.1146623839i\) |
\(L(1)\) |
\(\approx\) |
\(0.7145513908 - 0.04593634933i\) |
\(L(1)\) |
\(\approx\) |
\(0.7145513908 - 0.04593634933i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 + (-0.816 - 0.577i)T \) |
| 5 | \( 1 + (-0.552 + 0.833i)T \) |
| 7 | \( 1 + (0.969 + 0.243i)T \) |
| 11 | \( 1 + (-0.816 - 0.577i)T \) |
| 13 | \( 1 + (-0.696 + 0.717i)T \) |
| 17 | \( 1 + (0.602 - 0.798i)T \) |
| 19 | \( 1 + (0.739 + 0.673i)T \) |
| 23 | \( 1 + (-0.816 + 0.577i)T \) |
| 29 | \( 1 + (0.998 + 0.0615i)T \) |
| 31 | \( 1 + (-0.153 - 0.988i)T \) |
| 37 | \( 1 + (-0.850 + 0.526i)T \) |
| 41 | \( 1 + (0.998 - 0.0615i)T \) |
| 43 | \( 1 + (0.881 - 0.473i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.739 - 0.673i)T \) |
| 59 | \( 1 + (-0.969 + 0.243i)T \) |
| 61 | \( 1 + (-0.389 - 0.920i)T \) |
| 67 | \( 1 + (0.969 - 0.243i)T \) |
| 71 | \( 1 + (-0.445 + 0.895i)T \) |
| 73 | \( 1 + (0.445 - 0.895i)T \) |
| 79 | \( 1 + (-0.998 - 0.0615i)T \) |
| 83 | \( 1 + (-0.969 - 0.243i)T \) |
| 89 | \( 1 + (0.982 - 0.183i)T \) |
| 97 | \( 1 + (0.992 - 0.122i)T \) |
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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
−21.441154107549057505151114105729, −20.63564789318721654756559745710, −20.00597139181688627596697286325, −19.42480073162264539695638771838, −18.301932105099038696490556388836, −17.57207805373203698191988078064, −17.182547805177335251680740301709, −15.97987569000057480404042544515, −15.656254278741096115868194653705, −14.661392980336369166059600319853, −13.99457366111067817666509686144, −12.661491766757485254688658756708, −12.05359977706503219785573598738, −10.88982423958787338391313946389, −10.31057194480157935491654223068, −9.31345294308850846737820154256, −8.354586710681137803672317943975, −7.788168211151994408922533060587, −7.26430046111085688602250960042, −5.7990356767151661350649561875, −5.030189948748822649057587508663, −4.39512626803507097703946617944, −2.706359706011759184379745907364, −1.50626918520399639825129353585, −0.624928637565493979827192357109,
0.544401897773292995100122413333, 1.891322839186307110412593238846, 2.72708718172763348115718245142, 3.61435269674668579116126014424, 4.707156374431222773064738371777, 5.90909989843032677185430442749, 7.25089591444578501384298576172, 7.70392585709816566105176953286, 8.42056494013093925807661211555, 9.57003120716931701001526604646, 10.30398269041862918579761138454, 11.16763542360182850687952504873, 11.76981879983246749004419327119, 12.3145536614410057267845371980, 13.82183064253215588971177748509, 14.30630149034795548343847734511, 15.54790906962590895577648382518, 16.0623308523261972446485288817, 17.08919535178393603703460307369, 17.956935546992091677563143627045, 18.6108090770088173452723112489, 19.006284377392778986788991013156, 20.02086247570785550682029414000, 20.802619569721341340407928230112, 21.51376069669422350783498839598