| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 11-s − 13-s − 15-s − 17-s − 19-s + 21-s + 23-s + 25-s − 27-s − 29-s − 31-s + 33-s − 35-s − 37-s + 39-s + 41-s + 43-s + 45-s + 47-s + 49-s + 51-s − 53-s − 55-s + ⋯ |
| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 11-s − 13-s − 15-s − 17-s − 19-s + 21-s + 23-s + 25-s − 27-s − 29-s − 31-s + 33-s − 35-s − 37-s + 39-s + 41-s + 43-s + 45-s + 47-s + 49-s + 51-s − 53-s − 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 856 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 856 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7038450385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7038450385\) |
| \(L(1)\) |
\(\approx\) |
\(0.6442645579\) |
| \(L(1)\) |
\(\approx\) |
\(0.6442645579\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 107 | \( 1 \) |
| good | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 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
−22.05645270168107153036117376062, −21.27437978805747828099696512118, −20.47880933499774584506562094675, −19.235818457838139720437992878379, −18.67871842817893007130210228371, −17.66961281456379961355136120278, −17.19602763843263515024126292785, −16.422590747595537898630940719543, −15.61477867817811484303124136818, −14.75105964844771295088637799630, −13.45679934153755989426041750377, −12.8647641836046929001814493249, −12.43428494048814375083379407644, −10.94711037626755468040062597865, −10.576090743043904931637148848, −9.621466739496493000851883188283, −8.99638145084998511768768539504, −7.37884733144477432891396797836, −6.75766081479109013403732640223, −5.84360392149340000671020860587, −5.21999846072277285730719561983, −4.22457959404188806426440064400, −2.763966918551761783349191764945, −1.93054258424640892079651507759, −0.40225876930923032228253985738,
0.40225876930923032228253985738, 1.93054258424640892079651507759, 2.763966918551761783349191764945, 4.22457959404188806426440064400, 5.21999846072277285730719561983, 5.84360392149340000671020860587, 6.75766081479109013403732640223, 7.37884733144477432891396797836, 8.99638145084998511768768539504, 9.621466739496493000851883188283, 10.576090743043904931637148848, 10.94711037626755468040062597865, 12.43428494048814375083379407644, 12.8647641836046929001814493249, 13.45679934153755989426041750377, 14.75105964844771295088637799630, 15.61477867817811484303124136818, 16.422590747595537898630940719543, 17.19602763843263515024126292785, 17.66961281456379961355136120278, 18.67871842817893007130210228371, 19.235818457838139720437992878379, 20.47880933499774584506562094675, 21.27437978805747828099696512118, 22.05645270168107153036117376062
