L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 15-s + 16-s + 17-s + 18-s + 19-s + 20-s − 22-s − 24-s + 25-s − 26-s − 27-s + 29-s − 30-s − 31-s + 32-s + 33-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s − 11-s − 12-s − 13-s − 15-s + 16-s + 17-s + 18-s + 19-s + 20-s − 22-s − 24-s + 25-s − 26-s − 27-s + 29-s − 30-s − 31-s + 32-s + 33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.750625444\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.750625444\) |
\(L(1)\) |
\(\approx\) |
\(1.586762750\) |
\(L(1)\) |
\(\approx\) |
\(1.586762750\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 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
−28.226252687712368459295611300423, −26.77274857872284051945166964378, −25.56632477095717528263813972541, −24.6117817142226311236206426553, −23.79090883058220017624650656748, −22.85859667003473489391548494036, −21.93560577389607203213354983439, −21.3477563724487271676221702084, −20.36655620582330525564999594214, −18.80742558305517790053516488457, −17.717278743183782620940570904444, −16.741384089260543997842076207, −15.902790106330856173014191101230, −14.65074325152942269815807234764, −13.574027095976763595476127239559, −12.647676531253781502250337935563, −11.81400642901997425883153943088, −10.50030792309514014368953394528, −9.87029862675439222505852118235, −7.621130913731959828121210104193, −6.57503960231461631301215816788, −5.36486926249605990713385616200, −4.999082081045909074381129347452, −3.10218030541642144456326707603, −1.65723590715700437225096705131,
1.65723590715700437225096705131, 3.10218030541642144456326707603, 4.999082081045909074381129347452, 5.36486926249605990713385616200, 6.57503960231461631301215816788, 7.621130913731959828121210104193, 9.87029862675439222505852118235, 10.50030792309514014368953394528, 11.81400642901997425883153943088, 12.647676531253781502250337935563, 13.574027095976763595476127239559, 14.65074325152942269815807234764, 15.902790106330856173014191101230, 16.741384089260543997842076207, 17.717278743183782620940570904444, 18.80742558305517790053516488457, 20.36655620582330525564999594214, 21.3477563724487271676221702084, 21.93560577389607203213354983439, 22.85859667003473489391548494036, 23.79090883058220017624650656748, 24.6117817142226311236206426553, 25.56632477095717528263813972541, 26.77274857872284051945166964378, 28.226252687712368459295611300423