L(s) = 1 | + (−0.988 + 0.149i)2-s + (−0.826 − 0.563i)3-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.365 + 0.930i)9-s + (−0.0747 − 0.997i)10-s + (0.365 − 0.930i)11-s + (−0.955 − 0.294i)12-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)15-s + (0.826 − 0.563i)16-s + (0.733 − 0.680i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + (−0.988 + 0.149i)2-s + (−0.826 − 0.563i)3-s + (0.955 − 0.294i)4-s + (−0.0747 + 0.997i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)8-s + (0.365 + 0.930i)9-s + (−0.0747 − 0.997i)10-s + (0.365 − 0.930i)11-s + (−0.955 − 0.294i)12-s + (−0.623 − 0.781i)13-s + (0.623 − 0.781i)15-s + (0.826 − 0.563i)16-s + (0.733 − 0.680i)17-s + (−0.5 − 0.866i)18-s + (0.5 − 0.866i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4526374945 - 0.3649396927i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4526374945 - 0.3649396927i\) |
\(L(1)\) |
\(\approx\) |
\(0.5332979184 - 0.1017282858i\) |
\(L(1)\) |
\(\approx\) |
\(0.5332979184 - 0.1017282858i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (-0.988 + 0.149i)T \) |
| 3 | \( 1 + (-0.826 - 0.563i)T \) |
| 5 | \( 1 + (-0.0747 + 0.997i)T \) |
| 11 | \( 1 + (0.365 - 0.930i)T \) |
| 13 | \( 1 + (-0.623 - 0.781i)T \) |
| 17 | \( 1 + (0.733 - 0.680i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.733 - 0.680i)T \) |
| 29 | \( 1 + (-0.222 - 0.974i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.955 + 0.294i)T \) |
| 41 | \( 1 + (0.900 - 0.433i)T \) |
| 43 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (0.988 - 0.149i)T \) |
| 53 | \( 1 + (0.955 - 0.294i)T \) |
| 59 | \( 1 + (-0.0747 - 0.997i)T \) |
| 61 | \( 1 + (-0.955 - 0.294i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.222 + 0.974i)T \) |
| 73 | \( 1 + (0.988 + 0.149i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.623 + 0.781i)T \) |
| 89 | \( 1 + (-0.365 - 0.930i)T \) |
| 97 | \( 1 - T \) |
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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
−33.81673563514683632964833502994, −33.05540027285322622745576266029, −31.607980001208129352820864358752, −29.87798804536844025924147808263, −28.77962747920451630743785053817, −28.0383683469198228658006866716, −27.244490583313927299069584812705, −25.92765729252531665006231176550, −24.58497709080183659734125566438, −23.457709502767680430148608937549, −21.774430277321952624927913262099, −20.75790658628658318274808729945, −19.682777300469970552331404301138, −18.089853661694339894148606740815, −16.9440305267751633562005046288, −16.36905864873284539631381864573, −14.97949070631552482548006683371, −12.42111780073960012934846497126, −11.73826708443363893284862331561, −10.06125559903872400624944769196, −9.26633470280232120801367426568, −7.56429324856248070160313120345, −5.86024537608356784768719955870, −4.15260051906404845342471831553, −1.408585486354061211906783149012,
0.55994748654366490036597736261, 2.69442374460682321622145445074, 5.70735965166269706776401988443, 6.881279901881213725681474492565, 7.956986686952102998766231152754, 9.94747180657501843422216664260, 11.05515737358297221188709379746, 12.0181725750198338252462910458, 14.02126227293012270124399342984, 15.58736908400939826849537251926, 16.83283482842459197809184017565, 17.94859119417476904131241651892, 18.75908491162643972762062125610, 19.797682429616514590023757908655, 21.71563767876129177767051326878, 22.866111362017769199937239164983, 24.21670466719076062328618159778, 25.15878327367744369008123059169, 26.61253748374256932721496097459, 27.4439403651795772559763758470, 28.69947749859058258261246115205, 29.80665760024328432961035215962, 30.310205154609299272898627355096, 32.47129932748964562076157743388, 33.97399038148921317611032566564