L(s) = 1 | + (−0.986 − 0.164i)2-s + (0.180 − 0.983i)3-s + (0.945 + 0.324i)4-s + (−0.213 − 0.976i)5-s + (−0.340 + 0.940i)6-s + (0.922 − 0.386i)7-s + (−0.879 − 0.475i)8-s + (−0.934 − 0.355i)9-s + (0.0495 + 0.998i)10-s + (0.0495 − 0.998i)11-s + (0.490 − 0.871i)12-s + (0.371 + 0.928i)13-s + (−0.973 + 0.229i)14-s + (−0.999 + 0.0330i)15-s + (0.789 + 0.614i)16-s + (−0.277 − 0.960i)17-s + ⋯ |
L(s) = 1 | + (−0.986 − 0.164i)2-s + (0.180 − 0.983i)3-s + (0.945 + 0.324i)4-s + (−0.213 − 0.976i)5-s + (−0.340 + 0.940i)6-s + (0.922 − 0.386i)7-s + (−0.879 − 0.475i)8-s + (−0.934 − 0.355i)9-s + (0.0495 + 0.998i)10-s + (0.0495 − 0.998i)11-s + (0.490 − 0.871i)12-s + (0.371 + 0.928i)13-s + (−0.973 + 0.229i)14-s + (−0.999 + 0.0330i)15-s + (0.789 + 0.614i)16-s + (−0.277 − 0.960i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1021493201 - 0.8874939725i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1021493201 - 0.8874939725i\) |
\(L(1)\) |
\(\approx\) |
\(0.5666130916 - 0.5250309508i\) |
\(L(1)\) |
\(\approx\) |
\(0.5666130916 - 0.5250309508i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (-0.986 - 0.164i)T \) |
| 3 | \( 1 + (0.180 - 0.983i)T \) |
| 5 | \( 1 + (-0.213 - 0.976i)T \) |
| 7 | \( 1 + (0.922 - 0.386i)T \) |
| 11 | \( 1 + (0.0495 - 0.998i)T \) |
| 13 | \( 1 + (0.371 + 0.928i)T \) |
| 17 | \( 1 + (-0.277 - 0.960i)T \) |
| 19 | \( 1 + (0.991 - 0.131i)T \) |
| 23 | \( 1 + (-0.724 - 0.689i)T \) |
| 29 | \( 1 + (0.245 - 0.969i)T \) |
| 31 | \( 1 + (0.789 + 0.614i)T \) |
| 37 | \( 1 + (-0.768 - 0.639i)T \) |
| 41 | \( 1 + (-0.401 + 0.915i)T \) |
| 43 | \( 1 + (-0.627 - 0.778i)T \) |
| 47 | \( 1 + (0.789 + 0.614i)T \) |
| 53 | \( 1 + (0.894 + 0.446i)T \) |
| 59 | \( 1 + (-0.0825 - 0.996i)T \) |
| 61 | \( 1 + (-0.148 - 0.988i)T \) |
| 67 | \( 1 + (0.894 + 0.446i)T \) |
| 71 | \( 1 + (0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.768 - 0.639i)T \) |
| 79 | \( 1 + (0.601 + 0.799i)T \) |
| 83 | \( 1 + (-0.340 + 0.940i)T \) |
| 89 | \( 1 + (-0.340 + 0.940i)T \) |
| 97 | \( 1 + (-0.956 + 0.293i)T \) |
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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
−23.67473585995938028388709484172, −22.6848850437168431214656232258, −21.88398099892677914841606873684, −20.9796865024958400632250184506, −20.20368762902330531261308800573, −19.593490715138798057424217895688, −18.378750086135704482527995661950, −17.81175398737436509486167904120, −17.14650167205176639043175380312, −15.84995510616176911054292853666, −15.24652160337169795296507526386, −14.86484612578099356486472380484, −13.86535655361964030386162886858, −12.03268524043421722954198646270, −11.40747275471831759104994901645, −10.35161962419167126154834463122, −10.14621857685983203476189419033, −8.87534853608625502884892129288, −8.07756558205139921971600553828, −7.33236536090292002611414514443, −6.04380279249198247047287996970, −5.16974523340462800602589152674, −3.78187926009678820815966150685, −2.75622415095153535542681825316, −1.7080406627830659359465292045,
0.660207351431721701924233293488, 1.41534756053999416174964679269, 2.4886330494811145397830501092, 3.849295808911074092422068521088, 5.25687460607254607530867697942, 6.42603410975533709374095861568, 7.35891477440577266299321675291, 8.26472636008723614264274284629, 8.64921615025337089744364392808, 9.64126478592893010083390001665, 11.107199390628689102883918161223, 11.67626385289512299674287587908, 12.28460771705939826355534173465, 13.6752343729669487214973342047, 14.032008165511672316587349160600, 15.608220089945431513273691238024, 16.412911104866542905282261742658, 17.13066166031770573648621247562, 17.94628352779158916279225710508, 18.66011290214522153786829719535, 19.444886931890247458583490057008, 20.328673126841765136072867335498, 20.73117900169961986647938139059, 21.7179652779094210455506299446, 23.27988941142575258897055355766