L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + i·6-s + (0.866 − 0.5i)7-s + 8-s + (0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)12-s + (−0.5 − 0.866i)13-s + i·14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + (−0.866 + 0.5i)19-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 − 0.866i)4-s + i·6-s + (0.866 − 0.5i)7-s + 8-s + (0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)12-s + (−0.5 − 0.866i)13-s + i·14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + (−0.866 + 0.5i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.143936158 - 0.1707551985i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.143936158 - 0.1707551985i\) |
\(L(1)\) |
\(\approx\) |
\(1.070250697 + 0.01977063608i\) |
\(L(1)\) |
\(\approx\) |
\(1.070250697 + 0.01977063608i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + iT \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (-0.866 + 0.5i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.866 + 0.5i)T \) |
| 83 | \( 1 + (0.866 + 0.5i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 - T \) |
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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
−27.29045324650021072582669995215, −26.42185430523876243535052452051, −25.74557947600639328465264180662, −24.62174928354174131488419213716, −23.398814531903199798196932543533, −21.86801402154754768397687101175, −21.22437654742787140445566292323, −20.78904866796952595437445061696, −19.38854096426754306231687892870, −18.95863092786528455686499265446, −17.73770125709401753589352988388, −16.70633699230286969122104325504, −15.42415258014013427369164074215, −14.46320849372813840169652879181, −13.3618405478907828934498628084, −12.32615035490301768392629139845, −11.04642443668160554448259720803, −10.27080282485480884337430880878, −9.06715697225477067082634342377, −8.37037360605351078443009411525, −7.3854090395548158594346925540, −5.09182234484778669187827321958, −4.127675226152761324519338107922, −2.70393937214537821153822928356, −1.86759387521089751040814822580,
1.08990743883525153779600839099, 2.65575820101547528279647412, 4.45860355095487107433315400388, 5.64392218132957421089327902098, 7.24226443212672489247618374411, 7.7424200098522042578949201716, 8.66059180116940685445249083856, 9.88002649194558852444328238058, 10.88234807060005411072985346463, 12.68178701325670011733090098660, 13.61479302231860344589656473044, 14.6228097104482401759954085225, 15.18737259516915248287079266640, 16.48511820820594447897703720836, 17.63808963565696345201538576696, 18.329632531546609092001756973191, 19.24430766187661830630982719441, 20.33672241582305440401951340472, 21.08558294013987621214295826848, 22.85793367126899518691889838838, 23.67946419376025656348440327337, 24.47419575666070928618298388611, 25.28599765010391753946487395313, 26.083740970642334245845610139489, 27.08555706111500004271304141097