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 | \( 1 + (-0.984 - 0.173i)T \) |
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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
−28.42460180732454202116951917670, −27.72619300878607197908498906597, −26.69172784078516030782732831474, −25.847404663388427031479495793899, −24.72833691822663581422017856347, −24.0897085473471385180674821422, −23.106582806178749454963127809570, −21.831515621152833245176147547977, −20.98786947227301479950220566767, −19.20831753090720585466686785351, −18.65656972374548797366685823642, −17.58541142459657564498829274505, −16.46786851973783632256697523596, −15.67193934359933433570873968386, −14.44374340398332122917357855215, −13.78554663087240734283299779803, −12.19172713295242288877644378566, −10.93802780365636782253735865153, −9.55787224698573770787511373072, −8.518604179406631954496638865775, −7.66571452343185379834849267853, −6.10365802060042731161155589048, −5.36389818020839391230846898799, −3.78518411303727277730217927027, −1.60220040359184682775810604439,
1.09148185342715545172777845406, 2.645454293184059119413574013138, 4.07614686659135662776162647997, 5.18560342427939566233196297331, 7.31250537488351368702515845494, 8.160503219521121821387047394459, 9.62715830266033323101070078854, 10.48342409048485798253554047881, 11.508346741891468186409505052414, 12.63611339559711737832961219887, 13.63776759111201844511495421540, 14.75961263265470336781696766482, 16.30138948341427876962588678375, 17.57074452678582853126841097629, 18.0454665693668698739641753343, 19.375549612015988918416238760586, 20.61167010437365294260661274181, 20.68918495412025600279618776297, 22.359767851042419693757554444982, 22.99634661274386874200481340634, 24.25307446100330850350867451824, 25.69376705224180154924860867129, 26.45313820456515645635844578718, 27.67138504762267242881645250246, 28.076577031200062027530962254046