L(s) = 1 | − 2·7-s − 2·11-s − 2·17-s − 2·23-s − 25-s + 2·31-s − 2·41-s + 2·49-s + 2·53-s − 2·67-s + 2·71-s + 2·73-s + 4·77-s + 2·101-s + 2·103-s − 2·107-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + ⋯ |
L(s) = 1 | − 2·7-s − 2·11-s − 2·17-s − 2·23-s − 25-s + 2·31-s − 2·41-s + 2·49-s + 2·53-s − 2·67-s + 2·71-s + 2·73-s + 4·77-s + 2·101-s + 2·103-s − 2·107-s + 2·113-s + 4·119-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3089253058\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3089253058\) |
\(L(1)\) |
|
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
−10.19737043187140351711801789303, −9.528469047834042388493293917643, −9.009293671166468499404791861215, −8.480971055099012828025218778166, −8.326325336335636022341127485430, −7.80050111670424135503191945327, −7.43077438352719075513547102893, −6.87441862660598577907019260160, −6.50752955969908720603522920448, −6.13953809572977233238603293969, −5.99615877641375413605742674760, −5.08060148370656358236704242125, −5.07487114245196319272723291505, −4.22266232440364836589613707871, −3.89968880573234909674772514469, −3.36450347605110785160769639157, −2.77690535333401520316094638848, −2.34158363970689025362885140343, −2.02801588623418918675538848601, −0.39798184820396681645332112451,
0.39798184820396681645332112451, 2.02801588623418918675538848601, 2.34158363970689025362885140343, 2.77690535333401520316094638848, 3.36450347605110785160769639157, 3.89968880573234909674772514469, 4.22266232440364836589613707871, 5.07487114245196319272723291505, 5.08060148370656358236704242125, 5.99615877641375413605742674760, 6.13953809572977233238603293969, 6.50752955969908720603522920448, 6.87441862660598577907019260160, 7.43077438352719075513547102893, 7.80050111670424135503191945327, 8.326325336335636022341127485430, 8.480971055099012828025218778166, 9.009293671166468499404791861215, 9.528469047834042388493293917643, 10.19737043187140351711801789303