L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − i·18-s + (0.5 − 0.866i)19-s + 21-s + (−0.866 − 0.5i)22-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + (−0.866 − 0.5i)7-s − i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s + i·12-s − 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s − i·18-s + (0.5 − 0.866i)19-s + 21-s + (−0.866 − 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5125940037 - 1.197411113i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5125940037 - 1.197411113i\) |
\(L(1)\) |
\(\approx\) |
\(0.9672035042 - 0.5284296765i\) |
\(L(1)\) |
\(\approx\) |
\(0.9672035042 - 0.5284296765i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.866 + 0.5i)T \) |
| 47 | \( 1 + iT \) |
| 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 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 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
−32.4040394694906054545072453347, −31.18800384206185958771772651103, −30.29559409585395073876334685429, −29.04830207483297729126358287012, −28.36846031415455206013001769571, −26.53468694233219331768881067974, −25.32810327520158647846128483327, −24.4516858792258535267399649768, −23.24216553749734142307717447127, −22.56439825524119507631736664622, −21.60619197123922407907218078592, −20.07085097704010886747871817079, −18.50884120009137576675717316428, −17.36151821845176085914648038885, −16.185914863829118839444125129808, −15.29512661603756972716364939087, −13.58688598768088403777184008336, −12.608324538313098696729479980086, −11.812712147591674455953772185374, −10.16878857135937968702604511410, −8.04066587640755825574771008183, −6.68792939096067492965966139304, −5.80083277964727877438986423304, −4.38286064490164692859649165963, −2.37669733869627709197959212004,
0.55131570709625579190672329227, 3.07897389785157763310440644611, 4.4381278399772691596784137858, 5.76180648528618091931506071089, 6.87686194760521846777095654437, 9.48462288321028369132247231242, 10.61340197350599096022930282083, 11.54514672113834302323757498225, 12.870216786380335362194749782898, 13.89947603977856469114880928287, 15.67664769922084317592473860907, 16.19308186657015224925262964571, 17.87138629115403741686594601032, 19.333668023853324683813860656586, 20.48018022260096852075107419787, 21.7270400823043025013001667504, 22.4167550775094448675875818936, 23.47319027671087238623140951724, 24.3258201901021529221808856218, 26.11995150141457194065695508414, 27.312885182821131082733091449578, 28.70138421451477969421829907230, 29.16167414530608579029402995345, 30.23589740560603565069912317327, 31.761962941642766644353404116707