L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·5-s + i·8-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 25-s + (0.5 − 0.866i)29-s − i·31-s + (−0.866 + 0.5i)32-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + i·5-s + i·8-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.866 + 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 25-s + (0.5 − 0.866i)29-s − i·31-s + (−0.866 + 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.04035937202 + 2.090154684i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.04035937202 + 2.090154684i\) |
\(L(1)\) |
\(\approx\) |
\(1.142511047 + 0.9445516671i\) |
\(L(1)\) |
\(\approx\) |
\(1.142511047 + 0.9445516671i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + iT \) |
| 11 | \( 1 + (-0.866 - 0.5i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.866 - 0.5i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.866 + 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 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
−24.9305941250679068861687956184, −23.72220301028786219276669171570, −23.46768129756314232810163532144, −22.223615791973224833852115764, −21.27820967255589360805463267789, −20.55969492508469671516644711042, −19.91794246558245409001081937785, −18.80990579364443166330236154925, −17.737474239977562607813634505232, −16.36854299845150968589378551168, −15.76012917452043688250424686022, −14.622954011341790974488415108142, −13.63141952899847365427045737594, −12.684113091176534152696830954996, −12.19200042071693089345403596411, −10.918600752591499659722907598192, −9.97936929871990584890371483894, −8.88325195261076337524303318671, −7.566406369080737538440386694062, −6.2642801087330766222417268003, −5.039705938338958094110578623097, −4.52953252864850341941574356277, −3.0326297584357460780678097199, −1.83823391528615786593574860935, −0.43864108055828928597567190432,
2.15547301978563237417902420441, 3.22465978026250118491136957600, 4.210253822347712105638531200387, 5.710035421223637165254089135609, 6.30781994759845765386630441747, 7.59973988982645103252811489795, 8.24429202136998856386645770933, 10.03880643590580826008112156166, 10.97734640777054819955601207765, 11.92235332618066869546910925543, 13.0974961440521542742687976608, 13.857181506899931615903293673728, 14.881897580502660560966869950959, 15.45126835907709552914973062697, 16.57042210443212332302173523904, 17.52161667050207724113556083823, 18.59168748137642923493290292344, 19.51773396802065500439863846581, 20.90372564778847595170859937394, 21.55052594809824347500882117978, 22.35262568355725411756340833778, 23.39460781179878197035269359924, 23.78856952038852424594170716780, 25.08841745733967128570607184441, 25.8903189729649702934171832710