L(s) = 1 | + (0.458 − 0.888i)2-s + (−0.866 − 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 + 0.866i)9-s + (0.0950 + 0.995i)12-s + (−0.909 − 0.415i)13-s + (−0.327 + 0.945i)16-s + (0.690 + 0.723i)17-s + (0.998 − 0.0475i)18-s + (0.723 + 0.690i)19-s + (−0.945 − 0.327i)23-s + (0.928 + 0.371i)24-s + (−0.786 + 0.618i)26-s − i·27-s + ⋯ |
L(s) = 1 | + (0.458 − 0.888i)2-s + (−0.866 − 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 + 0.866i)9-s + (0.0950 + 0.995i)12-s + (−0.909 − 0.415i)13-s + (−0.327 + 0.945i)16-s + (0.690 + 0.723i)17-s + (0.998 − 0.0475i)18-s + (0.723 + 0.690i)19-s + (−0.945 − 0.327i)23-s + (0.928 + 0.371i)24-s + (−0.786 + 0.618i)26-s − i·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6402958240 - 0.7082850838i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6402958240 - 0.7082850838i\) |
\(L(1)\) |
\(\approx\) |
\(0.6619365495 - 0.4772068834i\) |
\(L(1)\) |
\(\approx\) |
\(0.6619365495 - 0.4772068834i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.458 - 0.888i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.909 - 0.415i)T \) |
| 17 | \( 1 + (0.690 + 0.723i)T \) |
| 19 | \( 1 + (0.723 + 0.690i)T \) |
| 23 | \( 1 + (-0.945 - 0.327i)T \) |
| 29 | \( 1 + (-0.959 + 0.281i)T \) |
| 31 | \( 1 + (-0.995 - 0.0950i)T \) |
| 37 | \( 1 + (-0.814 - 0.580i)T \) |
| 41 | \( 1 + (-0.841 + 0.540i)T \) |
| 43 | \( 1 + (-0.989 + 0.142i)T \) |
| 47 | \( 1 + (-0.998 - 0.0475i)T \) |
| 53 | \( 1 + (0.945 - 0.327i)T \) |
| 59 | \( 1 + (0.888 - 0.458i)T \) |
| 61 | \( 1 + (-0.0475 + 0.998i)T \) |
| 67 | \( 1 + (0.998 - 0.0475i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (-0.189 - 0.981i)T \) |
| 79 | \( 1 + (0.928 - 0.371i)T \) |
| 83 | \( 1 + (0.755 + 0.654i)T \) |
| 89 | \( 1 + (-0.235 - 0.971i)T \) |
| 97 | \( 1 + (-0.989 + 0.142i)T \) |
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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
−18.25431914742546331156181088629, −17.70958367645553027347709052599, −16.95947559428099499371615650006, −16.5297016669527456381156502602, −15.92171099815413802590137908062, −15.246638772743082318269316798866, −14.65433959866440634988225957250, −13.902746087391983857123187450843, −13.21890415332084236999352330734, −12.308755475025063568479996796431, −11.83972192024231036994731883037, −11.30101805627233912453857580917, −10.08016851586422167009677274137, −9.63361353287455442406298113907, −8.950149552261512514767414627175, −7.92734543734980569315250531317, −7.1063765421496164185575579019, −6.7900626056993707798471259575, −5.67713889008477595169098007200, −5.2740009454180949691472385344, −4.684722923891575988823469097162, −3.77140063745642525492835313919, −3.19252849265864725498782742170, −1.928185925551345899680190167724, −0.49089512161051816803327182799,
0.50501346564366714016268448064, 1.65263259288226987686588334901, 2.0198562762735187290806467451, 3.24028023588897692373075149223, 3.87577642268095907416447699305, 4.898606227695500081584307501495, 5.45305572285104971038253181513, 5.96933753098072233782165997993, 6.91674279583176826467096873300, 7.717932593084402108866343105309, 8.47397238036470250683263009925, 9.6328016148835193093248662718, 10.16759158474701957021929069116, 10.65067416429477992925764020072, 11.68376719138786653688707407096, 11.90173592611015933347304313917, 12.76453644555010115409542597559, 13.05102002666216623903626280300, 14.04486359214061758869458457015, 14.61758803958227395983668610717, 15.316404773516360608348537585521, 16.437714392525963379116013223661, 16.78626806529057281165989804957, 17.93059402142202150629906974009, 18.07427716170956958530621893681