L(s) = 1 | + 2·5-s − 2·7-s − 16-s + 2·17-s + 2·19-s − 2·23-s + 3·25-s − 4·35-s − 2·43-s + 2·47-s + 2·49-s − 2·73-s − 2·80-s − 81-s + 2·83-s + 4·85-s + 4·95-s + 2·112-s − 4·115-s − 4·119-s − 121-s + 4·125-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 2·5-s − 2·7-s − 16-s + 2·17-s + 2·19-s − 2·23-s + 3·25-s − 4·35-s − 2·43-s + 2·47-s + 2·49-s − 2·73-s − 2·80-s − 81-s + 2·83-s + 4·85-s + 4·95-s + 2·112-s − 4·115-s − 4·119-s − 121-s + 4·125-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.200010447\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.200010447\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04146358678350719389458911967, −9.750555883041708470138485995596, −9.703819952061487161337903822808, −9.433533239572184542543172121991, −8.763136772773211943346164645880, −8.500907774641200868953012118476, −7.68281396872433663174668837946, −7.25347571617745361085254469402, −7.00609858233756560852249127792, −6.22155327269194202381134855510, −6.15179120064968854027952070936, −5.81256104131291635026301797615, −5.25345105607793365622865073377, −4.98193484147774493703251283530, −3.99803428813241766683371529063, −3.47988804301851918412784710548, −3.00648139115225632376177120955, −2.61898833991937222704504863424, −1.86659777510316916573752527127, −1.12172049518387130788289792091,
1.12172049518387130788289792091, 1.86659777510316916573752527127, 2.61898833991937222704504863424, 3.00648139115225632376177120955, 3.47988804301851918412784710548, 3.99803428813241766683371529063, 4.98193484147774493703251283530, 5.25345105607793365622865073377, 5.81256104131291635026301797615, 6.15179120064968854027952070936, 6.22155327269194202381134855510, 7.00609858233756560852249127792, 7.25347571617745361085254469402, 7.68281396872433663174668837946, 8.500907774641200868953012118476, 8.763136772773211943346164645880, 9.433533239572184542543172121991, 9.703819952061487161337903822808, 9.750555883041708470138485995596, 10.04146358678350719389458911967