L(s) = 1 | + (0.245 + 0.969i)2-s + (−0.0825 + 0.996i)3-s + (−0.879 + 0.475i)4-s + (0.546 − 0.837i)5-s + (−0.986 + 0.164i)6-s + (−0.879 − 0.475i)7-s + (−0.677 − 0.735i)8-s + (−0.986 − 0.164i)9-s + (0.945 + 0.324i)10-s + (−0.986 + 0.164i)11-s + (−0.401 − 0.915i)12-s + (−0.0825 + 0.996i)13-s + (0.245 − 0.969i)14-s + (0.789 + 0.614i)15-s + (0.546 − 0.837i)16-s + (−0.879 − 0.475i)17-s + ⋯ |
L(s) = 1 | + (0.245 + 0.969i)2-s + (−0.0825 + 0.996i)3-s + (−0.879 + 0.475i)4-s + (0.546 − 0.837i)5-s + (−0.986 + 0.164i)6-s + (−0.879 − 0.475i)7-s + (−0.677 − 0.735i)8-s + (−0.986 − 0.164i)9-s + (0.945 + 0.324i)10-s + (−0.986 + 0.164i)11-s + (−0.401 − 0.915i)12-s + (−0.0825 + 0.996i)13-s + (0.245 − 0.969i)14-s + (0.789 + 0.614i)15-s + (0.546 − 0.837i)16-s + (−0.879 − 0.475i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.472 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1846889631 - 0.1105959234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1846889631 - 0.1105959234i\) |
\(L(1)\) |
\(\approx\) |
\(0.6005092305 + 0.3708618925i\) |
\(L(1)\) |
\(\approx\) |
\(0.6005092305 + 0.3708618925i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 \) |
good | 2 | \( 1 + (0.245 + 0.969i)T \) |
| 3 | \( 1 + (-0.0825 + 0.996i)T \) |
| 5 | \( 1 + (0.546 - 0.837i)T \) |
| 7 | \( 1 + (-0.879 - 0.475i)T \) |
| 11 | \( 1 + (-0.986 + 0.164i)T \) |
| 13 | \( 1 + (-0.0825 + 0.996i)T \) |
| 17 | \( 1 + (-0.879 - 0.475i)T \) |
| 23 | \( 1 + (-0.0825 - 0.996i)T \) |
| 29 | \( 1 + (-0.879 - 0.475i)T \) |
| 31 | \( 1 + (0.245 - 0.969i)T \) |
| 37 | \( 1 + (-0.986 + 0.164i)T \) |
| 41 | \( 1 + (0.789 + 0.614i)T \) |
| 43 | \( 1 + (0.945 - 0.324i)T \) |
| 47 | \( 1 + (-0.986 + 0.164i)T \) |
| 53 | \( 1 + (-0.986 + 0.164i)T \) |
| 59 | \( 1 + (0.789 + 0.614i)T \) |
| 61 | \( 1 + (-0.677 + 0.735i)T \) |
| 67 | \( 1 + (-0.677 - 0.735i)T \) |
| 71 | \( 1 + (-0.677 - 0.735i)T \) |
| 73 | \( 1 + (-0.879 - 0.475i)T \) |
| 79 | \( 1 + (0.945 - 0.324i)T \) |
| 83 | \( 1 + (0.546 + 0.837i)T \) |
| 89 | \( 1 + (-0.879 + 0.475i)T \) |
| 97 | \( 1 + (-0.677 + 0.735i)T \) |
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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
−24.828466972779373135456768448942, −23.77322137878594202823365632613, −22.871231321547685033689076528863, −22.33691159412366926525154186834, −21.50003912009043925809777582335, −20.3689917908259216675879243341, −19.342150209015724787843918055324, −18.930458066859228746271088832465, −17.831860625822321635145749148, −17.639219213664304308054926340953, −15.76393748937803676346136305837, −14.75109054157002438854383786207, −13.68483041557409490285922908564, −13.03492093948383757578582568762, −12.467293940597728674607387875056, −11.17995077558787984594688059242, −10.53040176797935580805437165526, −9.47858649722837265383922566777, −8.37262333172487221602468197424, −7.1205813122656977504409109055, −5.94934030072018267312262791813, −5.38263512943937931029581306374, −3.34624189043786431691655077414, −2.72012699470443226565181578534, −1.7488288472239645320280927712,
0.11366821147225390300174414561, 2.646114228085356216017245942048, 4.121937871775074092875069881100, 4.68973395578679788482842034805, 5.75267901611350229203990796292, 6.618905826247746755150378576727, 7.95623759412383647545766061931, 9.14188525784454499947142174667, 9.52695090799692282262041008954, 10.62069799936858132439800311860, 12.11639849280621570787081160386, 13.20837132887898806952750896631, 13.7596849183004758455159410297, 14.90285039936150151872788620422, 15.945346907455203580846304596470, 16.356819407145723505653607668319, 17.07882684611752313672029514983, 17.996604546574495070677049198339, 19.27602861442084095660407201838, 20.61977261857962809504816033386, 21.04367819881702673494386874868, 22.20876272287038299916308593672, 22.70449431647689079634867111409, 23.79313790474798881683580272844, 24.49580892525177369218047182102