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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0281 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0281 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6465839751 + 0.6286013576i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6465839751 + 0.6286013576i\) |
\(L(1)\) |
\(\approx\) |
\(0.7843434355 + 0.4429938623i\) |
\(L(1)\) |
\(\approx\) |
\(0.7843434355 + 0.4429938623i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
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
−28.076577031200062027530962254046, −27.67138504762267242881645250246, −26.45313820456515645635844578718, −25.69376705224180154924860867129, −24.25307446100330850350867451824, −22.99634661274386874200481340634, −22.359767851042419693757554444982, −20.68918495412025600279618776297, −20.61167010437365294260661274181, −19.375549612015988918416238760586, −18.0454665693668698739641753343, −17.57074452678582853126841097629, −16.30138948341427876962588678375, −14.75961263265470336781696766482, −13.63776759111201844511495421540, −12.63611339559711737832961219887, −11.508346741891468186409505052414, −10.48342409048485798253554047881, −9.62715830266033323101070078854, −8.160503219521121821387047394459, −7.31250537488351368702515845494, −5.18560342427939566233196297331, −4.07614686659135662776162647997, −2.645454293184059119413574013138, −1.09148185342715545172777845406,
1.60220040359184682775810604439, 3.78518411303727277730217927027, 5.36389818020839391230846898799, 6.10365802060042731161155589048, 7.66571452343185379834849267853, 8.518604179406631954496638865775, 9.55787224698573770787511373072, 10.93802780365636782253735865153, 12.19172713295242288877644378566, 13.78554663087240734283299779803, 14.44374340398332122917357855215, 15.67193934359933433570873968386, 16.46786851973783632256697523596, 17.58541142459657564498829274505, 18.65656972374548797366685823642, 19.20831753090720585466686785351, 20.98786947227301479950220566767, 21.831515621152833245176147547977, 23.106582806178749454963127809570, 24.0897085473471385180674821422, 24.72833691822663581422017856347, 25.847404663388427031479495793899, 26.69172784078516030782732831474, 27.72619300878607197908498906597, 28.42460180732454202116951917670