L(s) = 1 | + (−0.668 − 0.743i)3-s + (−0.925 − 0.378i)5-s + (0.372 − 0.928i)7-s + (−0.106 + 0.994i)9-s + (−0.831 − 0.554i)11-s + (0.549 + 0.835i)13-s + (0.337 + 0.941i)15-s + (0.630 + 0.776i)17-s + (−0.659 − 0.752i)19-s + (−0.939 + 0.343i)21-s + (0.915 + 0.401i)23-s + (0.713 + 0.700i)25-s + (0.810 − 0.585i)27-s + (−0.787 + 0.615i)29-s + (−0.858 + 0.512i)31-s + ⋯ |
L(s) = 1 | + (−0.668 − 0.743i)3-s + (−0.925 − 0.378i)5-s + (0.372 − 0.928i)7-s + (−0.106 + 0.994i)9-s + (−0.831 − 0.554i)11-s + (0.549 + 0.835i)13-s + (0.337 + 0.941i)15-s + (0.630 + 0.776i)17-s + (−0.659 − 0.752i)19-s + (−0.939 + 0.343i)21-s + (0.915 + 0.401i)23-s + (0.713 + 0.700i)25-s + (0.810 − 0.585i)27-s + (−0.787 + 0.615i)29-s + (−0.858 + 0.512i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3092421526 - 0.6734280906i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3092421526 - 0.6734280906i\) |
\(L(1)\) |
\(\approx\) |
\(0.6429741077 - 0.2675019043i\) |
\(L(1)\) |
\(\approx\) |
\(0.6429741077 - 0.2675019043i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 \) |
good | 3 | \( 1 + (-0.668 - 0.743i)T \) |
| 5 | \( 1 + (-0.925 - 0.378i)T \) |
| 7 | \( 1 + (0.372 - 0.928i)T \) |
| 11 | \( 1 + (-0.831 - 0.554i)T \) |
| 13 | \( 1 + (0.549 + 0.835i)T \) |
| 17 | \( 1 + (0.630 + 0.776i)T \) |
| 19 | \( 1 + (-0.659 - 0.752i)T \) |
| 23 | \( 1 + (0.915 + 0.401i)T \) |
| 29 | \( 1 + (-0.787 + 0.615i)T \) |
| 31 | \( 1 + (-0.858 + 0.512i)T \) |
| 37 | \( 1 + (-0.845 + 0.533i)T \) |
| 41 | \( 1 + (-0.997 - 0.0750i)T \) |
| 43 | \( 1 + (0.889 + 0.457i)T \) |
| 47 | \( 1 + (-0.360 - 0.932i)T \) |
| 53 | \( 1 + (0.659 - 0.752i)T \) |
| 59 | \( 1 + (0.998 - 0.0500i)T \) |
| 61 | \( 1 + (-0.143 + 0.989i)T \) |
| 67 | \( 1 + (0.337 - 0.941i)T \) |
| 71 | \( 1 + (0.289 - 0.957i)T \) |
| 73 | \( 1 + (0.764 + 0.644i)T \) |
| 79 | \( 1 + (0.229 - 0.973i)T \) |
| 83 | \( 1 + (-0.958 + 0.283i)T \) |
| 89 | \( 1 + (0.349 - 0.937i)T \) |
| 97 | \( 1 + (0.429 + 0.902i)T \) |
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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
−18.51392693988365682822881986697, −18.23481724776065316026146249582, −17.316694715351928293542176996830, −16.60245093881935087694529651928, −15.750617002996712720527464213701, −15.48879063584400883818141318370, −14.87096646050258030787455168624, −14.30137202067100716608854149891, −12.883843824316055804196070675364, −12.50267384383545210702805873317, −11.7500788201926268380591449842, −11.108722984854844995243421611839, −10.60541954568121287424138207538, −9.87614107558143649541210252469, −9.0094784225557863014437231535, −8.271388551362929987839021161048, −7.59388372619708020319663837712, −6.790549740199823468359517292909, −5.71466681897400610225071670358, −5.379525686047404587438418737172, −4.54838457881247930457765708724, −3.731402640756714290487602012781, −3.03718777850365571320897891054, −2.16558706584778582842693402307, −0.75916362479408867762035931195,
0.34243621336422134187866376162, 1.230489835215030486242299760238, 1.934751001157008472211351208314, 3.29902199013305337846736911261, 3.89512835785669238279954226763, 4.92543418076318773635141077455, 5.292692284812016719604370362, 6.417851319660136411527068070794, 7.11153130690015404494615013023, 7.58328382079621153386431544302, 8.419931548509512761083019993340, 8.85483012867376466403939089011, 10.281672494923532889767255219957, 10.87683281494333580485883039041, 11.298546108179377951354797769156, 11.96016003962456572968219008767, 12.99084669513192192846260864317, 13.1143640320234380408840381511, 14.0009983321229968645540963297, 14.828042544445551503619517239933, 15.64343731995709517064146624608, 16.52839297831680185619657924199, 16.7081049003230060278679619457, 17.456425317722444848281282061584, 18.33784969886118510283291609577