L(s) = 1 | + (0.995 − 0.0950i)2-s + (0.5 − 0.866i)3-s + (0.981 − 0.189i)4-s + (−0.235 − 0.971i)5-s + (0.415 − 0.909i)6-s + (0.959 − 0.281i)8-s + (−0.5 − 0.866i)9-s + (−0.327 − 0.945i)10-s + (0.327 − 0.945i)12-s + (−0.654 − 0.755i)13-s + (−0.959 − 0.281i)15-s + (0.928 − 0.371i)16-s + (−0.888 + 0.458i)17-s + (−0.580 − 0.814i)18-s + (−0.888 − 0.458i)19-s + (−0.415 − 0.909i)20-s + ⋯ |
L(s) = 1 | + (0.995 − 0.0950i)2-s + (0.5 − 0.866i)3-s + (0.981 − 0.189i)4-s + (−0.235 − 0.971i)5-s + (0.415 − 0.909i)6-s + (0.959 − 0.281i)8-s + (−0.5 − 0.866i)9-s + (−0.327 − 0.945i)10-s + (0.327 − 0.945i)12-s + (−0.654 − 0.755i)13-s + (−0.959 − 0.281i)15-s + (0.928 − 0.371i)16-s + (−0.888 + 0.458i)17-s + (−0.580 − 0.814i)18-s + (−0.888 − 0.458i)19-s + (−0.415 − 0.909i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.107265086 - 2.637451448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107265086 - 2.637451448i\) |
\(L(1)\) |
\(\approx\) |
\(1.617521736 - 1.210703811i\) |
\(L(1)\) |
\(\approx\) |
\(1.617521736 - 1.210703811i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.995 - 0.0950i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.235 - 0.971i)T \) |
| 13 | \( 1 + (-0.654 - 0.755i)T \) |
| 17 | \( 1 + (-0.888 + 0.458i)T \) |
| 19 | \( 1 + (-0.888 - 0.458i)T \) |
| 23 | \( 1 + (0.928 - 0.371i)T \) |
| 29 | \( 1 + (-0.841 + 0.540i)T \) |
| 31 | \( 1 + (0.327 + 0.945i)T \) |
| 37 | \( 1 + (0.981 + 0.189i)T \) |
| 41 | \( 1 + (0.415 - 0.909i)T \) |
| 43 | \( 1 + (0.959 - 0.281i)T \) |
| 47 | \( 1 + (-0.580 + 0.814i)T \) |
| 53 | \( 1 + (0.928 + 0.371i)T \) |
| 59 | \( 1 + (0.995 + 0.0950i)T \) |
| 61 | \( 1 + (0.580 - 0.814i)T \) |
| 67 | \( 1 + (0.580 + 0.814i)T \) |
| 71 | \( 1 + (0.841 - 0.540i)T \) |
| 73 | \( 1 + (-0.786 - 0.618i)T \) |
| 79 | \( 1 + (-0.235 - 0.971i)T \) |
| 83 | \( 1 + (-0.142 - 0.989i)T \) |
| 89 | \( 1 + (-0.0475 + 0.998i)T \) |
| 97 | \( 1 + (0.959 - 0.281i)T \) |
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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
−22.59633905267534004946201278342, −21.44535109906236562810462157777, −21.27078932507339956257417577509, −20.10364646053021496658477921217, −19.47634685519631399094069265557, −18.72633111664305811441202883405, −17.29955857303363001062416056665, −16.54620146532430205367342106874, −15.69381838175421529689239890522, −14.85119230062501127225573643330, −14.67493691003088187000666279824, −13.669007250044656374946097156375, −12.94068996922606331528706107648, −11.43267372075043333105362075248, −11.31569566465367177463562094092, −10.22115909748058917655599106430, −9.40329827138751740106601686621, −8.147030156200756926005554358900, −7.2791943750309845056860102525, −6.45880667921629693733497730094, −5.41077269662819400012225617447, −4.30935040426622626794310977014, −3.88128308627068950001456949979, −2.6507699314379055837600469705, −2.22716781339788440673737973151,
0.81237483158017336173238263596, 1.96948311335617405580578902792, 2.797269149730135708151348896339, 3.920879416260855609320834559788, 4.81994674504402566070075465596, 5.7358798591819632753415341442, 6.73008905487008805751968954728, 7.51353677278975791909481198318, 8.428170864145358327259706518158, 9.22817560878214424106105576217, 10.60085255055121403401110736169, 11.48773400593146485107474161303, 12.51775943468491276251093760585, 12.82553055287207331482082452138, 13.43090322708786535547234672413, 14.55019848368787753995665109417, 15.111924589785079677328412342344, 15.97298716428952308479020829434, 17.06588013250552221750861655525, 17.6378746661532186735243394362, 19.06412808334422799422448303794, 19.59894020780432687295403215497, 20.26159705726595261131027071797, 20.87425197756615242562694551048, 21.79958248963228591423767686917