L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)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 − i·20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 29-s + (−0.866 + 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 − 0.5i)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 − i·20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 29-s + (−0.866 + 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.348 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.454247137 + 2.093058862i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.454247137 + 2.093058862i\) |
\(L(1)\) |
\(\approx\) |
\(1.411938446 + 0.6740996381i\) |
\(L(1)\) |
\(\approx\) |
\(1.411938446 + 0.6740996381i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 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 | \( 1 - T \) |
| 31 | \( 1 + (-0.866 + 0.5i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 - iT \) |
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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
−24.92306629835473557970238616735, −24.133717861290278330742337108960, −23.163731242496370124709918241434, −22.48299952292543696718392005700, −21.86955470130160294896794398691, −20.36683177810678831957337493911, −20.09472339517230657456363155752, −18.93039868994649423367824174032, −18.22431960889808421305320240432, −16.606725855052670793038974951733, −15.70851077807211546785556181893, −14.746856316994906898752687193162, −14.15933064390296467457221847964, −12.908096797429185039316976557710, −11.86783167984822139455697800044, −11.400485505213053071550403335979, −10.22579876524987818019483795241, −9.184467409674709870984538930312, −7.506085893482914197377644270186, −6.77889118982676174600961371606, −5.45803623277626925192808556447, −4.261963699493745736501530455600, −3.44594625284467310455066416445, −2.23038232316506599691782105988, −0.61341549491801727622292576147,
1.434963316334812713947708747875, 3.36765586785877650537667968564, 3.96425216472325039928712624278, 5.22815184309601377616695461512, 6.19450053530742172546015614338, 7.47731349711212030415248555460, 8.18498553483109022641514398895, 9.348285919675821343010134272172, 11.07105560068593007712964343843, 11.85982967175317809896941705502, 12.6242163706334891142541989236, 13.69771800748445101717459831676, 14.66889714325206079430468240314, 15.48005083199891927663172539255, 16.51740686178269736367632690685, 16.95198911220197117418616562237, 18.41493538965173015880691503403, 19.69667359242229994255969630388, 20.27823582638515295922445359475, 21.4597410253234850770321247570, 22.198208825523253970718414628685, 23.22858281714879676880167020356, 23.88560221172559941332931258756, 24.65996926364321613841517382729, 25.50238301697962945686910617767