L(s) = 1 | + (−0.945 − 0.324i)3-s + (−0.401 + 0.915i)5-s − 7-s + (0.789 + 0.614i)9-s + (0.0825 + 0.996i)11-s + (0.945 − 0.324i)13-s + (0.677 − 0.735i)15-s + (−0.879 − 0.475i)17-s + (0.986 + 0.164i)19-s + (0.945 + 0.324i)21-s + (0.986 + 0.164i)23-s + (−0.677 − 0.735i)25-s + (−0.546 − 0.837i)27-s + (−0.677 + 0.735i)29-s + (−0.789 − 0.614i)31-s + ⋯ |
L(s) = 1 | + (−0.945 − 0.324i)3-s + (−0.401 + 0.915i)5-s − 7-s + (0.789 + 0.614i)9-s + (0.0825 + 0.996i)11-s + (0.945 − 0.324i)13-s + (0.677 − 0.735i)15-s + (−0.879 − 0.475i)17-s + (0.986 + 0.164i)19-s + (0.945 + 0.324i)21-s + (0.986 + 0.164i)23-s + (−0.677 − 0.735i)25-s + (−0.546 − 0.837i)27-s + (−0.677 + 0.735i)29-s + (−0.789 − 0.614i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9019899685 + 0.4139505816i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9019899685 + 0.4139505816i\) |
\(L(1)\) |
\(\approx\) |
\(0.6945900895 + 0.08826667780i\) |
\(L(1)\) |
\(\approx\) |
\(0.6945900895 + 0.08826667780i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 191 | \( 1 \) |
good | 3 | \( 1 + (-0.945 - 0.324i)T \) |
| 5 | \( 1 + (-0.401 + 0.915i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.0825 + 0.996i)T \) |
| 13 | \( 1 + (0.945 - 0.324i)T \) |
| 17 | \( 1 + (-0.879 - 0.475i)T \) |
| 19 | \( 1 + (0.986 + 0.164i)T \) |
| 23 | \( 1 + (0.986 + 0.164i)T \) |
| 29 | \( 1 + (-0.677 + 0.735i)T \) |
| 31 | \( 1 + (-0.789 - 0.614i)T \) |
| 37 | \( 1 + (0.789 - 0.614i)T \) |
| 41 | \( 1 + (0.546 - 0.837i)T \) |
| 43 | \( 1 + (-0.245 - 0.969i)T \) |
| 47 | \( 1 + (0.0825 + 0.996i)T \) |
| 53 | \( 1 + (-0.0825 - 0.996i)T \) |
| 59 | \( 1 + (-0.789 - 0.614i)T \) |
| 61 | \( 1 + (-0.879 + 0.475i)T \) |
| 67 | \( 1 + (0.879 - 0.475i)T \) |
| 71 | \( 1 + (-0.546 + 0.837i)T \) |
| 73 | \( 1 + (-0.0825 + 0.996i)T \) |
| 79 | \( 1 + (0.677 + 0.735i)T \) |
| 83 | \( 1 + (0.986 - 0.164i)T \) |
| 89 | \( 1 + (0.245 - 0.969i)T \) |
| 97 | \( 1 + (0.789 - 0.614i)T \) |
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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
−22.0348198892642713970345618329, −21.45853687993775687163916331274, −20.51339714747002850819545934730, −19.70033279605716855652730719214, −18.82169954977273652225095375369, −18.043808458028045695807013735174, −16.850001253961876036890451847019, −16.482763913837292433252207678651, −15.84862271710509791021320484666, −15.11255848056785394976316632603, −13.42742621164407638986277288411, −13.14439419270176004947152992732, −12.0979360978074718633171394320, −11.32059178518724026674392695269, −10.68327441345882619203248471909, −9.3505651979193630279959201209, −9.012392580342701323810105121850, −7.746702644204667220999575497455, −6.51749104358224761361211714658, −5.95392626689379197194944065535, −4.96051375624316835842542902774, −3.99993037927835266670614796451, −3.22063182312752598998756933965, −1.31169889645132572135548479959, −0.47216543870397745187818546291,
0.6260679819572992749191405917, 2.04771890465586005301863393258, 3.23850099229394045808999552913, 4.16076183366996930100189109709, 5.397561596940454247459568072121, 6.27042348401435369289822827031, 7.097774480835511627233105756424, 7.50616141541998302546731677997, 9.12125290868092135578681770843, 9.96273790960128267732028956430, 10.92348957234883900788724115091, 11.38017471218922266812092024063, 12.496844606480353970053691418945, 13.057914465812811767770717353238, 14.028419979974025619684148345692, 15.27578936391621338916695311848, 15.77799222512274472509704837364, 16.58061758491789411426801190770, 17.64736048644000823537108760159, 18.28869503215442284413660850284, 18.8434144406526069772421802290, 19.80881162707267704470630103565, 20.59217282536592003230720355368, 21.95783993116302990450075283358, 22.479211244361864206696794607845