L(s) = 1 | + (0.866 + 0.5i)7-s + (0.5 − 0.866i)11-s + (−0.866 + 0.5i)13-s + i·17-s + 19-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + i·37-s + (−0.5 − 0.866i)41-s + (−0.866 − 0.5i)43-s + (0.866 + 0.5i)47-s + (0.5 + 0.866i)49-s − i·53-s + (−0.5 − 0.866i)59-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)7-s + (0.5 − 0.866i)11-s + (−0.866 + 0.5i)13-s + i·17-s + 19-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)29-s + (0.5 + 0.866i)31-s + i·37-s + (−0.5 − 0.866i)41-s + (−0.866 − 0.5i)43-s + (0.866 + 0.5i)47-s + (0.5 + 0.866i)49-s − i·53-s + (−0.5 − 0.866i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.247755886 + 0.1280349004i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247755886 + 0.1280349004i\) |
\(L(1)\) |
\(\approx\) |
\(1.150986172 + 0.05959578971i\) |
\(L(1)\) |
\(\approx\) |
\(1.150986172 + 0.05959578971i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.866 - 0.5i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−27.22771322860579569026642120947, −26.5764899653348907553985003552, −25.12921000100150618710592635461, −24.62476246734739236479766362351, −23.38721471365444536881892806797, −22.59921242669407409566315450580, −21.512709689455368933943225761256, −20.35819810099196498787229094777, −19.86552752368321613307578594176, −18.36357177813481566268674875746, −17.56767727278390769160093184471, −16.71612653323519125823425108364, −15.35823047763477375555177677187, −14.50715165356162013150812081661, −13.5519805549268831853831644002, −12.23245635101436593321553098545, −11.38324877820371172065999807808, −10.145516313153380818033451828686, −9.18243120433944807714777578975, −7.6919560562418092639218574579, −7.053715268595070701248963009130, −5.298164261750483729179292395214, −4.45451512805452230808117089894, −2.86754378419040266770519535812, −1.310965258979160927966616342991,
1.47586334913344436097842806534, 2.946230958257883919928305688857, 4.45650180577082903029479704477, 5.557436671457912371943283984149, 6.816163457465083306024955766068, 8.15216140452578814082156319984, 8.99120349647003621593024144176, 10.317778473229664438196156174889, 11.51101748600717609656928257101, 12.20600045961437085917739417211, 13.67547030146616364717412176415, 14.53289333873309017614442598311, 15.46864140809056656798117821173, 16.76351039481201343855014409969, 17.51972048337208855081006529801, 18.736643631878831096042798773998, 19.47506883173669823489302691693, 20.75265193789697195350628721735, 21.63092835106601760956877061262, 22.35510618211326776795703072099, 23.79384013148157202758740663034, 24.4410239425095904137226965703, 25.26415296859714508518474842213, 26.73670531682748079558044569692, 27.10891496006995461662308281786