L(s) = 1 | + (−0.691 + 0.722i)2-s + (0.753 + 0.657i)3-s + (−0.0448 − 0.998i)4-s + (0.858 − 0.512i)5-s + (−0.995 + 0.0896i)6-s + (0.753 + 0.657i)8-s + (0.134 + 0.990i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)12-s + (−0.691 + 0.722i)13-s + (0.983 + 0.178i)15-s + (−0.995 + 0.0896i)16-s + (0.936 + 0.351i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (−0.550 − 0.834i)20-s + ⋯ |
L(s) = 1 | + (−0.691 + 0.722i)2-s + (0.753 + 0.657i)3-s + (−0.0448 − 0.998i)4-s + (0.858 − 0.512i)5-s + (−0.995 + 0.0896i)6-s + (0.753 + 0.657i)8-s + (0.134 + 0.990i)9-s + (−0.222 + 0.974i)10-s + (0.623 − 0.781i)12-s + (−0.691 + 0.722i)13-s + (0.983 + 0.178i)15-s + (−0.995 + 0.0896i)16-s + (0.936 + 0.351i)17-s + (−0.809 − 0.587i)18-s + (−0.809 + 0.587i)19-s + (−0.550 − 0.834i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8923463948 + 1.077901006i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8923463948 + 1.077901006i\) |
\(L(1)\) |
\(\approx\) |
\(0.9438549110 + 0.5631966804i\) |
\(L(1)\) |
\(\approx\) |
\(0.9438549110 + 0.5631966804i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.691 + 0.722i)T \) |
| 3 | \( 1 + (0.753 + 0.657i)T \) |
| 5 | \( 1 + (0.858 - 0.512i)T \) |
| 13 | \( 1 + (-0.691 + 0.722i)T \) |
| 17 | \( 1 + (0.936 + 0.351i)T \) |
| 19 | \( 1 + (-0.809 + 0.587i)T \) |
| 23 | \( 1 + (0.623 + 0.781i)T \) |
| 29 | \( 1 + (-0.963 + 0.266i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.963 + 0.266i)T \) |
| 41 | \( 1 + (0.753 + 0.657i)T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.983 - 0.178i)T \) |
| 53 | \( 1 + (0.936 - 0.351i)T \) |
| 59 | \( 1 + (0.753 - 0.657i)T \) |
| 61 | \( 1 + (0.936 + 0.351i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.550 + 0.834i)T \) |
| 73 | \( 1 + (0.983 + 0.178i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.691 - 0.722i)T \) |
| 89 | \( 1 + (-0.900 + 0.433i)T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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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
−22.98763858023448757557701029335, −22.19350461758377520010816184962, −21.14738999491075228894683578423, −20.71210262458374361633627582378, −19.677166151362350483526161271564, −18.9815284876420150297615468470, −18.34209002635549035122682957950, −17.49533642910600180486278038912, −16.92379186999367839610457077677, −15.4268216654204195863991576, −14.47369832664809079594264646239, −13.65566919033262121908282103516, −12.77653402486905310028186992957, −12.18191836021423633487474936191, −10.84786400711888130164370058782, −10.11528559502748828742594373950, −9.243852601925350783213630279389, −8.4821043488695707464306486434, −7.37072641945071094680524183657, −6.80448222152030650171860984508, −5.36884099732443703408762487849, −3.74989468539960299217879785535, −2.71844860047005535010816461842, −2.1912620604624667350739486351, −0.89566744611192273476495687531,
1.48162462271723499886935501325, 2.3824337430380395031935460045, 4.00873048460834886686481851325, 5.07201411645229427065939410387, 5.82191769793648869355065310146, 7.073480543236864144842895068673, 8.07244745515643853422765484204, 8.86836455579289877194030872188, 9.68893731822663953527380295773, 10.098090635599493823972902634667, 11.2648559416127679810852018582, 12.79206197297866930560123283643, 13.73666739543956905617832078186, 14.53653831731806904439776083179, 15.07219412222256852106661964933, 16.278436997083700984023222249565, 16.804821907762365175868901438862, 17.458709632490597783392014499625, 18.79537096520925178516928610274, 19.26475695668340407088811995123, 20.31853246920527227790991796232, 21.07119540192723582027999438721, 21.78243258869677999472591193365, 22.91512014352292365992312516299, 24.01921551270751476009087827122