L(s) = 1 | + (−0.856 − 0.515i)2-s + (−0.370 + 0.928i)3-s + (0.468 + 0.883i)4-s + (−0.994 − 0.108i)5-s + (0.796 − 0.605i)6-s + (−0.947 − 0.319i)7-s + (0.0541 − 0.998i)8-s + (−0.725 − 0.687i)9-s + (0.796 + 0.605i)10-s + (−0.561 − 0.827i)11-s + (−0.994 + 0.108i)12-s + (−0.725 + 0.687i)13-s + (0.647 + 0.762i)14-s + (0.468 − 0.883i)15-s + (−0.561 + 0.827i)16-s + (−0.947 + 0.319i)17-s + ⋯ |
L(s) = 1 | + (−0.856 − 0.515i)2-s + (−0.370 + 0.928i)3-s + (0.468 + 0.883i)4-s + (−0.994 − 0.108i)5-s + (0.796 − 0.605i)6-s + (−0.947 − 0.319i)7-s + (0.0541 − 0.998i)8-s + (−0.725 − 0.687i)9-s + (0.796 + 0.605i)10-s + (−0.561 − 0.827i)11-s + (−0.994 + 0.108i)12-s + (−0.725 + 0.687i)13-s + (0.647 + 0.762i)14-s + (0.468 − 0.883i)15-s + (−0.561 + 0.827i)16-s + (−0.947 + 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.0004992210914 + 0.03215327025i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0004992210914 + 0.03215327025i\) |
\(L(1)\) |
\(\approx\) |
\(0.3316177767 + 0.02308879173i\) |
\(L(1)\) |
\(\approx\) |
\(0.3316177767 + 0.02308879173i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (-0.856 - 0.515i)T \) |
| 3 | \( 1 + (-0.370 + 0.928i)T \) |
| 5 | \( 1 + (-0.994 - 0.108i)T \) |
| 7 | \( 1 + (-0.947 - 0.319i)T \) |
| 11 | \( 1 + (-0.561 - 0.827i)T \) |
| 13 | \( 1 + (-0.725 + 0.687i)T \) |
| 17 | \( 1 + (-0.947 + 0.319i)T \) |
| 19 | \( 1 + (-0.161 + 0.986i)T \) |
| 23 | \( 1 + (0.267 - 0.963i)T \) |
| 29 | \( 1 + (-0.856 + 0.515i)T \) |
| 31 | \( 1 + (-0.161 - 0.986i)T \) |
| 37 | \( 1 + (0.0541 + 0.998i)T \) |
| 41 | \( 1 + (0.267 + 0.963i)T \) |
| 43 | \( 1 + (-0.561 + 0.827i)T \) |
| 47 | \( 1 + (-0.994 + 0.108i)T \) |
| 53 | \( 1 + (0.796 - 0.605i)T \) |
| 61 | \( 1 + (-0.856 - 0.515i)T \) |
| 67 | \( 1 + (0.0541 - 0.998i)T \) |
| 71 | \( 1 + (-0.994 + 0.108i)T \) |
| 73 | \( 1 + (0.647 + 0.762i)T \) |
| 79 | \( 1 + (-0.370 - 0.928i)T \) |
| 83 | \( 1 + (0.907 + 0.419i)T \) |
| 89 | \( 1 + (-0.856 + 0.515i)T \) |
| 97 | \( 1 + (0.647 - 0.762i)T \) |
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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
−32.192202023681935166576706264481, −31.05364832045977246869775732830, −29.679249638215407981636338173788, −28.65501129256885376083679385250, −27.8434355113590320971662219015, −26.48094312247498375689976105946, −25.43089314429587255688514738975, −24.38776401594500979048063184350, −23.33638391757830074836446683343, −22.51060498671244058868639125193, −19.94533788790444590001107100698, −19.47463435084816700440323766366, −18.29299098317296604304553744857, −17.34292438507883417701231077721, −15.9213453393103502767474443731, −15.09920727874262454036763364669, −13.151568482007144488829088461186, −11.931120675425586191549831427386, −10.647515045757100436900804867704, −9.02359088925234431837890783204, −7.535489335422812287671596555126, −6.90083495263581498235577154970, −5.27159898837186661074764008919, −2.51255355890100796187765340625, −0.05165411900596720614647235625,
3.11991320000170495507024578230, 4.28612756241541179007995266592, 6.55681694493927345072790421214, 8.20575634992234592380777762500, 9.4480796803001560987847359329, 10.628552681589691909430402689022, 11.58114111818600009927695026520, 12.85841983283730429581749808814, 15.08837832658481974532451076598, 16.40956571337656722606627174044, 16.66643822501373817458042447846, 18.564520954106162122583012682275, 19.5976686104267744831184297689, 20.55800516385480493679141078490, 21.82040561197918729360563918406, 22.81596698281219581909110657636, 24.281464132835949995142262847954, 26.2097271346278434635932833901, 26.62113082891310420429265920693, 27.62865295484397132012909263060, 28.76575697067933057131213170914, 29.42876910059118630200510909496, 31.19866926233840896484328366610, 31.981055114290506390008476892071, 33.49152363861265135618303866075