| L(s) = 1 | + (0.642 − 0.766i)5-s + (0.5 − 0.866i)7-s − i·11-s + (−0.642 − 0.766i)13-s + (0.766 + 0.642i)17-s + (0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.342 + 0.939i)29-s − 31-s + (−0.342 − 0.939i)35-s − i·37-s + (0.173 − 0.984i)41-s + (0.342 + 0.939i)43-s + (−0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯ |
| L(s) = 1 | + (0.642 − 0.766i)5-s + (0.5 − 0.866i)7-s − i·11-s + (−0.642 − 0.766i)13-s + (0.766 + 0.642i)17-s + (0.939 + 0.342i)23-s + (−0.173 − 0.984i)25-s + (−0.342 + 0.939i)29-s − 31-s + (−0.342 − 0.939i)35-s − i·37-s + (0.173 − 0.984i)41-s + (0.342 + 0.939i)43-s + (−0.939 − 0.342i)47-s + (−0.5 − 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1809661772 - 1.432614180i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(-0.1809661772 - 1.432614180i\) |
| \(L(1)\) |
\(\approx\) |
\(1.047989604 - 0.4607116982i\) |
| \(L(1)\) |
\(\approx\) |
\(1.047989604 - 0.4607116982i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + (0.642 - 0.766i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-0.642 - 0.766i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.342 + 0.939i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.342 + 0.939i)T \) |
| 47 | \( 1 + (-0.939 - 0.342i)T \) |
| 53 | \( 1 + (-0.984 + 0.173i)T \) |
| 59 | \( 1 + (-0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.642 + 0.766i)T \) |
| 67 | \( 1 + (0.984 - 0.173i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (-0.766 - 0.642i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−19.26426308702731991343950466524, −18.62533683201750434733215711268, −18.188181164113813604787117293367, −17.37703640072138581013063451068, −16.859363444774839190774063449828, −15.811465722553670450816138792443, −15.06046192349191590967546176433, −14.47036114965005716361538180733, −14.12263599643132773381010068665, −12.92508073177963751063554985408, −12.41218044370886226457368056894, −11.465801734791795523435576586073, −11.05315337487061202229082318546, −9.79431757052908025642812289572, −9.636591146953474661992033082380, −8.75050048484564786484236126854, −7.627448144581061609685162122064, −7.12433047002882610989200680925, −6.30124821677606684605912933159, −5.39472606784540423226347380430, −4.87256956519941571276699372689, −3.80497497147810545556552414817, −2.64182603834400431682172901586, −2.239398557210732617629870767251, −1.35659335454926641725264117281,
0.22070891196161282889976909139, 1.10403340177750871893650163538, 1.71692069480985873684036427392, 3.015700283505629727574916989467, 3.69575530860355806104574546299, 4.8237538628466024675474451997, 5.30635056089521807804487354750, 6.06420870615237545767107735630, 7.07578204993042792326625991622, 7.90045819879496511397864121184, 8.469565988663644979733499875004, 9.33970545064893046578214228981, 10.08034257161497277936959556363, 10.78480658038475242864093340826, 11.413914969447667983561069268089, 12.57924089003728954657665043916, 12.91751591795216219761228455673, 13.75913574723788976481470074012, 14.3657886037789661635175874184, 15.03215787497064451980736479254, 16.12757553326514904759538021018, 16.73735668929953489422196530106, 17.1926621148976319917667603153, 17.815672919827670678007258840528, 18.70681750883313150451438306300
