L(s) = 1 | + (0.281 − 0.959i)2-s + (0.755 + 0.654i)3-s + (−0.841 − 0.540i)4-s + (0.841 − 0.540i)6-s + (−0.909 + 0.415i)7-s + (−0.755 + 0.654i)8-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.281 − 0.959i)12-s + (−0.909 − 0.415i)13-s + (0.142 + 0.989i)14-s + (0.415 + 0.909i)16-s + (0.540 + 0.841i)17-s + (0.989 + 0.142i)18-s + (−0.841 − 0.540i)19-s + ⋯ |
L(s) = 1 | + (0.281 − 0.959i)2-s + (0.755 + 0.654i)3-s + (−0.841 − 0.540i)4-s + (0.841 − 0.540i)6-s + (−0.909 + 0.415i)7-s + (−0.755 + 0.654i)8-s + (0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (−0.281 − 0.959i)12-s + (−0.909 − 0.415i)13-s + (0.142 + 0.989i)14-s + (0.415 + 0.909i)16-s + (0.540 + 0.841i)17-s + (0.989 + 0.142i)18-s + (−0.841 − 0.540i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4098728619 + 0.5585969550i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4098728619 + 0.5585969550i\) |
\(L(1)\) |
\(\approx\) |
\(0.9392126136 - 0.08461386273i\) |
\(L(1)\) |
\(\approx\) |
\(0.9392126136 - 0.08461386273i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.281 - 0.959i)T \) |
| 3 | \( 1 + (0.755 + 0.654i)T \) |
| 7 | \( 1 + (-0.909 + 0.415i)T \) |
| 11 | \( 1 + (-0.959 + 0.281i)T \) |
| 13 | \( 1 + (-0.909 - 0.415i)T \) |
| 17 | \( 1 + (0.540 + 0.841i)T \) |
| 19 | \( 1 + (-0.841 - 0.540i)T \) |
| 29 | \( 1 + (-0.841 + 0.540i)T \) |
| 31 | \( 1 + (-0.654 - 0.755i)T \) |
| 37 | \( 1 + (0.989 - 0.142i)T \) |
| 41 | \( 1 + (-0.142 + 0.989i)T \) |
| 43 | \( 1 + (0.755 + 0.654i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.909 - 0.415i)T \) |
| 59 | \( 1 + (-0.415 + 0.909i)T \) |
| 61 | \( 1 + (-0.654 - 0.755i)T \) |
| 67 | \( 1 + (0.281 - 0.959i)T \) |
| 71 | \( 1 + (-0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.540 - 0.841i)T \) |
| 79 | \( 1 + (-0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.989 + 0.142i)T \) |
| 89 | \( 1 + (0.654 - 0.755i)T \) |
| 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
−29.04338543796135012065822637152, −27.2251174313858232339832438984, −26.346005809080629131828543558738, −25.64208550426462906964444535108, −24.741449254376235100416697544694, −23.70886890035678038927676642534, −23.01783525290992147306815439875, −21.67223791572738846795082543060, −20.49501080234296327386494067045, −19.131650119488305925593956521363, −18.45821314645765238588308715414, −17.08693125055655494790236078731, −16.101273982391942660174480250308, −14.94551615992978673632666011881, −13.95231854881730831865448262824, −13.08957493690968793715517527243, −12.23457642735406772388337938559, −10.00856194441828093103515375515, −8.94824167583951976480632472526, −7.66448511272171889248321100044, −6.96432603477314954533652740110, −5.64640007028111385085380571605, −3.95338953463660073117938212064, −2.68876876088831000044918288147, −0.22390554121599431317014388325,
2.26869606032832739691529199833, 3.15758070713358036095612906034, 4.47422169969439434176578907317, 5.715494251856968465732692149240, 7.84681478551477540917972440118, 9.182705411520856774098314634752, 9.98027411976213007842178350523, 10.90726650013486896671225508407, 12.641838273730668037301338978163, 13.14564552594562762053600572555, 14.67912123163407785203818887423, 15.30958401838329613599404831070, 16.76158278425550838794729135755, 18.35672382389142746941561031670, 19.37356051934716646707412819811, 20.02747531517911695629349195311, 21.19430929522024363038471368820, 21.89099523340548938901349351476, 22.840202207337071654267271815989, 24.1117987209843667277304561616, 25.62085189941271521662202627224, 26.30919503612146746910703613626, 27.53034795635205928147940960651, 28.30017680403326750205872063802, 29.35397979159761595724678622053