L(s) = 1 | + (−0.654 − 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (−0.142 − 0.989i)6-s + (0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)10-s + (0.654 − 0.755i)11-s + (−0.654 + 0.755i)12-s + (−0.959 + 0.281i)13-s + (−0.415 − 0.909i)14-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.142 + 0.989i)17-s + ⋯ |
L(s) = 1 | + (−0.654 − 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (−0.142 − 0.989i)6-s + (0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (0.959 − 0.281i)10-s + (0.654 − 0.755i)11-s + (−0.654 + 0.755i)12-s + (−0.959 + 0.281i)13-s + (−0.415 − 0.909i)14-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.142 + 0.989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.193390215 + 0.2513170686i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.193390215 + 0.2513170686i\) |
\(L(1)\) |
\(\approx\) |
\(1.024776888 + 0.06836387700i\) |
\(L(1)\) |
\(\approx\) |
\(1.024776888 + 0.06836387700i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.654 - 0.755i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 11 | \( 1 + (0.654 - 0.755i)T \) |
| 13 | \( 1 + (-0.959 + 0.281i)T \) |
| 17 | \( 1 + (0.142 + 0.989i)T \) |
| 19 | \( 1 + (0.142 - 0.989i)T \) |
| 29 | \( 1 + (-0.142 - 0.989i)T \) |
| 31 | \( 1 + (0.841 - 0.540i)T \) |
| 37 | \( 1 + (-0.415 - 0.909i)T \) |
| 41 | \( 1 + (0.415 - 0.909i)T \) |
| 43 | \( 1 + (-0.841 - 0.540i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.959 + 0.281i)T \) |
| 59 | \( 1 + (-0.959 + 0.281i)T \) |
| 61 | \( 1 + (-0.841 + 0.540i)T \) |
| 67 | \( 1 + (0.654 + 0.755i)T \) |
| 71 | \( 1 + (-0.654 - 0.755i)T \) |
| 73 | \( 1 + (-0.142 + 0.989i)T \) |
| 79 | \( 1 + (0.959 - 0.281i)T \) |
| 83 | \( 1 + (-0.415 - 0.909i)T \) |
| 89 | \( 1 + (-0.841 - 0.540i)T \) |
| 97 | \( 1 + (-0.415 + 0.909i)T \) |
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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
−38.199345952938279834194250800475, −36.85880858660055521816147706964, −36.15326235998923348993679305748, −35.13458641489215004260488820128, −33.560340825986917411840731135035, −32.202160011262934899636313295659, −31.143738407768407904012382129503, −29.44533162742435653149063788306, −27.70102760915420388331662960571, −26.86628733966501129913588702712, −25.085663704080201694499223054965, −24.53571898560322130309389782054, −23.274507660989997587087547744148, −20.539451934818267454102740931592, −19.725151936370181535441078356765, −18.14111882842997627545344585960, −16.874925313993120683214553437949, −15.140311256924691708540367847441, −14.04445956117997645153264879685, −12.11256698180227244551608080116, −9.63430524416145883796502627340, −8.26915988698542440107253097219, −7.24691052521206073122623024527, −4.76819409837205579260399461408, −1.3797846159960658038612160980,
2.422807249193995525545659390855, 4.06368017958575994735722742671, 7.57658983559309863798267647923, 8.8795589013945002698155122045, 10.489122351861352389188803100170, 11.70549832767387652110164846562, 13.96328593621444567575480909438, 15.28336328858085073315065112200, 17.20644616703920879527303097668, 18.89371584858930917798322621504, 19.71961408220323290488564258838, 21.3261356712269760072526249366, 22.11537274776449995636983231261, 24.55984974550267442921257070393, 26.21775611958856808744246734444, 26.951390997049195127006939735059, 27.988419931107052543334527769683, 30.01430159849003592791241125303, 30.74581808560397652685458492907, 31.97348917347789808407302969171, 33.93016426705979391415339165970, 35.04956605322416709788179171930, 36.85009267046127835910036362665, 37.46216404834207596121828809078, 38.52346429453796222459047680066