L(s) = 1 | + (−0.430 + 0.902i)2-s + (−0.956 + 0.292i)3-s + (−0.630 − 0.776i)4-s + (0.889 + 0.456i)5-s + (0.147 − 0.989i)6-s + (0.263 − 0.964i)7-s + (0.972 − 0.234i)8-s + (0.829 − 0.558i)9-s + (−0.794 + 0.606i)10-s + (0.937 + 0.348i)11-s + (0.829 + 0.558i)12-s + (−0.998 + 0.0592i)13-s + (0.757 + 0.652i)14-s + (−0.984 − 0.176i)15-s + (−0.205 + 0.978i)16-s + (−0.320 + 0.947i)17-s + ⋯ |
L(s) = 1 | + (−0.430 + 0.902i)2-s + (−0.956 + 0.292i)3-s + (−0.630 − 0.776i)4-s + (0.889 + 0.456i)5-s + (0.147 − 0.989i)6-s + (0.263 − 0.964i)7-s + (0.972 − 0.234i)8-s + (0.829 − 0.558i)9-s + (−0.794 + 0.606i)10-s + (0.937 + 0.348i)11-s + (0.829 + 0.558i)12-s + (−0.998 + 0.0592i)13-s + (0.757 + 0.652i)14-s + (−0.984 − 0.176i)15-s + (−0.205 + 0.978i)16-s + (−0.320 + 0.947i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5878487496 + 0.4161656256i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5878487496 + 0.4161656256i\) |
\(L(1)\) |
\(\approx\) |
\(0.6798902140 + 0.3335542338i\) |
\(L(1)\) |
\(\approx\) |
\(0.6798902140 + 0.3335542338i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 107 | \( 1 \) |
good | 2 | \( 1 + (-0.430 + 0.902i)T \) |
| 3 | \( 1 + (-0.956 + 0.292i)T \) |
| 5 | \( 1 + (0.889 + 0.456i)T \) |
| 7 | \( 1 + (0.263 - 0.964i)T \) |
| 11 | \( 1 + (0.937 + 0.348i)T \) |
| 13 | \( 1 + (-0.998 + 0.0592i)T \) |
| 17 | \( 1 + (-0.320 + 0.947i)T \) |
| 19 | \( 1 + (0.992 + 0.118i)T \) |
| 23 | \( 1 + (0.757 - 0.652i)T \) |
| 29 | \( 1 + (-0.0887 + 0.996i)T \) |
| 31 | \( 1 + (-0.533 - 0.845i)T \) |
| 37 | \( 1 + (0.375 + 0.926i)T \) |
| 41 | \( 1 + (0.482 - 0.875i)T \) |
| 43 | \( 1 + (0.889 - 0.456i)T \) |
| 47 | \( 1 + (0.482 + 0.875i)T \) |
| 53 | \( 1 + (-0.430 - 0.902i)T \) |
| 59 | \( 1 + (-0.0887 - 0.996i)T \) |
| 61 | \( 1 + (0.263 + 0.964i)T \) |
| 67 | \( 1 + (0.972 + 0.234i)T \) |
| 71 | \( 1 + (-0.956 - 0.292i)T \) |
| 73 | \( 1 + (-0.717 - 0.696i)T \) |
| 79 | \( 1 + (-0.915 + 0.403i)T \) |
| 83 | \( 1 + (-0.861 + 0.508i)T \) |
| 89 | \( 1 + (0.147 + 0.989i)T \) |
| 97 | \( 1 + (-0.794 + 0.606i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.23461742603054088560607315868, −28.59981614413501094042285236267, −27.671132087700510720055156173741, −26.809868837127558901314326929246, −24.98298490766237596022746336011, −24.64401516484030613085861142076, −22.825222852006284550698844539417, −21.894323614615946550611811819, −21.4240236521615277855604722999, −19.9891991041294337254249965403, −18.80649433648249338681454454503, −17.83667014437612584028423809337, −17.22123004373885130387820525428, −16.1217104237720961311117034688, −14.14064835073313950953136110453, −12.94729597214855433555859585250, −11.9964011113020380332518523281, −11.26954814920514185843388269364, −9.73910232027026015519473556058, −9.04925515038774609844603782591, −7.33591343756439927517888812127, −5.68672382553171341715455645961, −4.71560249562422398606003842566, −2.542189088152522593996921856774, −1.22127061519362479953674042217,
1.36280321863161345966842563105, 4.21394862948259348157115491145, 5.38388673930203729829917920808, 6.59159697451269578673221292778, 7.3112633676988767628715986996, 9.29421577630795950293624576669, 10.14260947276003265487807371256, 11.05886715044781640314920837995, 12.81801229217351456872542899055, 14.204894518821306369699022322845, 14.95310057104193295014633682393, 16.511952551114600108996400736823, 17.23714071365070902494443318869, 17.73227292142372023710843417990, 19.026369979253085300585902708359, 20.46701411451596537739413076623, 22.12904632109889323591259380089, 22.48758172520201883112089039092, 23.82710899811254714669089400020, 24.57517902019308549315595334105, 25.868109851687619267429789441212, 26.82062628064702416185949807463, 27.49004618134333557107388055615, 28.78095132889219527044784787668, 29.48557324700791613763025686509