L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 4·29-s + 6·32-s − 8·58-s + 7·64-s − 12·116-s + 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 5·16-s − 4·29-s + 6·32-s − 8·58-s + 7·64-s − 12·116-s + 2·121-s + 127-s + 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.594417780\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.594417780\) |
\(L(1)\) |
|
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
−9.659383482076321123103911949970, −9.507238998594422910190845815944, −8.845207762284210828230873090792, −8.409820620381303945906461342835, −7.71028414437749712951653289369, −7.60420758735182749836449887121, −7.22004980049350719028832459650, −6.86504228984014016252219503405, −6.18460956923895546934513110706, −6.07407141785245440799488904403, −5.46804349956058881904074425448, −5.31451908067224196334220458268, −4.79943829362728001219307872881, −4.25645906281573239712506438275, −3.74845511281811893891380091675, −3.63332251567808974695681433885, −3.01196616588470107968316419511, −2.36291571661188020919983003193, −1.94540061496984898281872094842, −1.39157937932907991579790493048,
1.39157937932907991579790493048, 1.94540061496984898281872094842, 2.36291571661188020919983003193, 3.01196616588470107968316419511, 3.63332251567808974695681433885, 3.74845511281811893891380091675, 4.25645906281573239712506438275, 4.79943829362728001219307872881, 5.31451908067224196334220458268, 5.46804349956058881904074425448, 6.07407141785245440799488904403, 6.18460956923895546934513110706, 6.86504228984014016252219503405, 7.22004980049350719028832459650, 7.60420758735182749836449887121, 7.71028414437749712951653289369, 8.409820620381303945906461342835, 8.845207762284210828230873090792, 9.507238998594422910190845815944, 9.659383482076321123103911949970