L(s) = 1 | + (0.717 + 0.696i)2-s + (0.586 − 0.809i)3-s + (0.0310 + 0.999i)4-s + (0.664 − 0.747i)5-s + (0.984 − 0.172i)6-s + (−0.514 − 0.857i)7-s + (−0.673 + 0.739i)8-s + (−0.311 − 0.950i)9-s + (0.997 − 0.0744i)10-s + (−0.535 − 0.844i)11-s + (0.827 + 0.561i)12-s + (−0.977 + 0.209i)13-s + (0.227 − 0.973i)14-s + (−0.215 − 0.976i)15-s + (−0.998 + 0.0620i)16-s + (0.798 + 0.601i)17-s + ⋯ |
L(s) = 1 | + (0.717 + 0.696i)2-s + (0.586 − 0.809i)3-s + (0.0310 + 0.999i)4-s + (0.664 − 0.747i)5-s + (0.984 − 0.172i)6-s + (−0.514 − 0.857i)7-s + (−0.673 + 0.739i)8-s + (−0.311 − 0.950i)9-s + (0.997 − 0.0744i)10-s + (−0.535 − 0.844i)11-s + (0.827 + 0.561i)12-s + (−0.977 + 0.209i)13-s + (0.227 − 0.973i)14-s + (−0.215 − 0.976i)15-s + (−0.998 + 0.0620i)16-s + (0.798 + 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.618 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.067192174 - 1.003375419i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.067192174 - 1.003375419i\) |
\(L(1)\) |
\(\approx\) |
\(1.738422934 - 0.2447841633i\) |
\(L(1)\) |
\(\approx\) |
\(1.738422934 - 0.2447841633i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.717 + 0.696i)T \) |
| 3 | \( 1 + (0.586 - 0.809i)T \) |
| 5 | \( 1 + (0.664 - 0.747i)T \) |
| 7 | \( 1 + (-0.514 - 0.857i)T \) |
| 11 | \( 1 + (-0.535 - 0.844i)T \) |
| 13 | \( 1 + (-0.977 + 0.209i)T \) |
| 17 | \( 1 + (0.798 + 0.601i)T \) |
| 19 | \( 1 + (0.931 - 0.363i)T \) |
| 29 | \( 1 + (0.545 - 0.837i)T \) |
| 31 | \( 1 + (0.992 + 0.123i)T \) |
| 37 | \( 1 + (0.700 - 0.713i)T \) |
| 41 | \( 1 + (0.179 - 0.983i)T \) |
| 43 | \( 1 + (-0.691 + 0.722i)T \) |
| 47 | \( 1 + (-0.917 + 0.398i)T \) |
| 53 | \( 1 + (0.997 + 0.0744i)T \) |
| 59 | \( 1 + (-0.885 + 0.465i)T \) |
| 61 | \( 1 + (-0.847 + 0.530i)T \) |
| 67 | \( 1 + (-0.287 + 0.957i)T \) |
| 71 | \( 1 + (0.948 - 0.317i)T \) |
| 73 | \( 1 + (0.545 + 0.837i)T \) |
| 79 | \( 1 + (-0.944 - 0.329i)T \) |
| 83 | \( 1 + (0.955 + 0.293i)T \) |
| 89 | \( 1 + (0.783 - 0.621i)T \) |
| 97 | \( 1 + (0.969 + 0.245i)T \) |
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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
−23.04900941650068621867714787881, −22.59642924705434926483319244260, −21.774255736857303120524815369544, −21.326937953115778656134708719409, −20.36884066109185601149056472198, −19.66921132254868178452137805953, −18.66023289987666814908036020028, −18.09149937035279392102070157500, −16.602541678572577583275113159275, −15.49664675419246378128159977772, −14.984790655901957937676773621562, −14.23930996387475274524677176149, −13.44505767251514265291097319326, −12.413327745462348256703184107527, −11.57848947995425061895745228987, −10.26730157355413759598500063955, −9.90782898642917867111076031509, −9.292917905163542607672502829760, −7.738604062898595065048019501764, −6.50823135052215164763462525431, −5.32857995120776664575470025887, −4.85368988168120399700839131698, −3.2266335084071377474561056368, −2.85234910482861782864230643038, −1.92780634162574991417009204472,
0.883015831790137363871578209049, 2.456594590414486594404338814130, 3.33906847531956292932287545378, 4.53162885524559750961518702444, 5.669247255290998544968243398215, 6.42221385043289143796534487178, 7.48178458636291313126286650452, 8.10114434297987641324196590666, 9.14723708110467907176193349941, 10.094411466767009441081104263438, 11.73089589733739132018044384648, 12.55663514898196848316618298830, 13.29946540089315791286640845697, 13.81801841938925041882030002348, 14.4934464551805422975707319937, 15.72922792521477200159553327131, 16.62750364936002878338925364113, 17.24548038384663339651728044893, 18.08452821761835211074303125331, 19.329223547331248804307214974079, 20.04716887487527864809832647891, 21.034494194449996256088460490757, 21.558989884535049811306261113506, 22.81799848339337604191866974237, 23.54498626173192409038877167379