L(s) = 1 | + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.642 + 0.766i)7-s + (−0.866 − 0.5i)8-s + (−0.173 + 0.984i)11-s + (−0.342 + 0.939i)13-s + (−0.939 − 0.342i)14-s + (0.173 − 0.984i)16-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.984 + 0.173i)22-s + (0.642 + 0.766i)23-s − 26-s − i·28-s + (−0.939 + 0.342i)29-s + ⋯ |
L(s) = 1 | + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−0.642 + 0.766i)7-s + (−0.866 − 0.5i)8-s + (−0.173 + 0.984i)11-s + (−0.342 + 0.939i)13-s + (−0.939 − 0.342i)14-s + (0.173 − 0.984i)16-s + (−0.866 + 0.5i)17-s + (0.5 − 0.866i)19-s + (−0.984 + 0.173i)22-s + (0.642 + 0.766i)23-s − 26-s − i·28-s + (−0.939 + 0.342i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1973844825 + 0.8922157794i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1973844825 + 0.8922157794i\) |
\(L(1)\) |
\(\approx\) |
\(0.6930336152 + 0.6818125796i\) |
\(L(1)\) |
\(\approx\) |
\(0.6930336152 + 0.6818125796i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 7 | \( 1 + (-0.642 + 0.766i)T \) |
| 11 | \( 1 + (-0.173 + 0.984i)T \) |
| 13 | \( 1 + (-0.342 + 0.939i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.642 + 0.766i)T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 + (0.766 - 0.642i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.939 + 0.342i)T \) |
| 43 | \( 1 + (-0.984 - 0.173i)T \) |
| 47 | \( 1 + (0.642 - 0.766i)T \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.866 + 0.5i)T \) |
| 79 | \( 1 + (0.939 - 0.342i)T \) |
| 83 | \( 1 + (-0.342 - 0.939i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.984 + 0.173i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−28.425537389916296799815788089283, −26.96209340685445623830148353307, −26.715605565275343628204798478, −24.98550592959109201577154202546, −23.9443682081654148021020482776, −22.78172993793198573866101628203, −22.28865920582610270246666202781, −20.9430454733194807634924337218, −20.162322397428994110671562176451, −19.24087689512029113865491264004, −18.27277191561083298973496266423, −17.03431419679198673790630428133, −15.79290128384278963935131242645, −14.42596193985375356618423860830, −13.43573944299247595101148630969, −12.68791022870275549548899598406, −11.32343869212864969363949382411, −10.43790909106912229958131826777, −9.44673838169132533348231881958, −8.07068175184698295424235594514, −6.40397856321229516234843701042, −5.113978857216060316700911665113, −3.72606707881584073486959138782, −2.71569854450107075993197536942, −0.74639189987299175817090813827,
2.50009448261397231982032689316, 4.11311403704549026633088447978, 5.27654828541576169798020858634, 6.52716861955908358973577044244, 7.41999605598337358016666724461, 8.9473758208444360367400689187, 9.61818404438130566490341660714, 11.56659159925344499044292127639, 12.6933701882470698373317556538, 13.51951731694876195436988991296, 14.95244460267215491470490294720, 15.500870911848199167675375965896, 16.65373112657646002420601903463, 17.669324191701350277293129445, 18.657739676482360606194081202151, 19.84477133939342334720593479749, 21.39235799887369940028887629504, 22.12424314015951311431086735798, 23.06395826673924762049109378777, 24.100307164905234973104940123156, 24.992411569285387460945149670564, 25.95306250187922847634678064493, 26.59234252982295838776396823156, 28.019689865781559014957102244529, 28.75664054124077973889998398598