L(s) = 1 | − 16-s + 4·19-s + 4·49-s − 4·61-s − 4·109-s + 127-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 + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
L(s) = 1 | − 16-s + 4·19-s + 4·49-s − 4·61-s − 4·109-s + 127-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 + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8045466795\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8045466795\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.62033086590413236406189592231, −7.45237123227378816983632751686, −7.32067924072005349420657053653, −7.13931642202132727536844916255, −6.85744546489755928801315468096, −6.55672195386414276608856355551, −6.36693237660249894171426637406, −5.94177936700939655715355141630, −5.74710953651620593703347171559, −5.69949412272248130018087159535, −5.34522420386373569232417709751, −4.96641370577638982848358959965, −4.95770460266557193651478883978, −4.69447148435520395844182353373, −4.32768987079081907197032935784, −3.76971133479913460119449386306, −3.74322955803488103635970183538, −3.71103122058905116286250460304, −3.01702673398197661037316773425, −2.83457145538099995172097371699, −2.51040726453255872723452292787, −2.50138346045873189197283043621, −1.65089928933564993302833260922, −1.24946640468036301038987573849, −1.13445124268816426143868428178,
1.13445124268816426143868428178, 1.24946640468036301038987573849, 1.65089928933564993302833260922, 2.50138346045873189197283043621, 2.51040726453255872723452292787, 2.83457145538099995172097371699, 3.01702673398197661037316773425, 3.71103122058905116286250460304, 3.74322955803488103635970183538, 3.76971133479913460119449386306, 4.32768987079081907197032935784, 4.69447148435520395844182353373, 4.95770460266557193651478883978, 4.96641370577638982848358959965, 5.34522420386373569232417709751, 5.69949412272248130018087159535, 5.74710953651620593703347171559, 5.94177936700939655715355141630, 6.36693237660249894171426637406, 6.55672195386414276608856355551, 6.85744546489755928801315468096, 7.13931642202132727536844916255, 7.32067924072005349420657053653, 7.45237123227378816983632751686, 7.62033086590413236406189592231