L(s) = 1 | + 4·3-s + 10·9-s − 2·16-s − 4·17-s + 20·27-s − 8·48-s − 2·49-s − 16·51-s − 4·53-s − 4·61-s + 4·71-s + 4·79-s + 34·81-s − 4·83-s + 127-s + 131-s + 137-s + 139-s − 20·144-s − 8·147-s + 149-s + 151-s − 40·153-s + 157-s − 16·159-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 4·3-s + 10·9-s − 2·16-s − 4·17-s + 20·27-s − 8·48-s − 2·49-s − 16·51-s − 4·53-s − 4·61-s + 4·71-s + 4·79-s + 34·81-s − 4·83-s + 127-s + 131-s + 137-s + 139-s − 20·144-s − 8·147-s + 149-s + 151-s − 40·153-s + 157-s − 16·159-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{4} \cdot 47^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.986613724\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.986613724\) |
\(L(1)\) |
|
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
−7.76807304728872810474613486327, −7.37889227873632408428905342381, −7.32592420002085790698621608933, −6.81267419863250867388740139920, −6.80543112905942585348483167052, −6.72170602454147341278508062384, −6.42239838291363031223512022359, −6.23078548815443055948851362510, −6.08019481742297483042924822901, −5.08739090476776641076545251497, −4.91331005623260117295370210204, −4.79876274307482899583782422644, −4.56935911595939851380123199415, −4.46965917503732683491478775816, −4.11512688321308378648967089916, −3.94593363704164899158115402591, −3.57063600973940416759050567102, −3.35970391375422249321562918347, −2.90150301558418022506540329796, −2.81681354362779972551415009849, −2.49529307446799026054557928504, −2.27441389998862881456962322526, −1.84847062897443005881527195053, −1.74253400638481136706800819564, −1.52141779790946051256389488615,
1.52141779790946051256389488615, 1.74253400638481136706800819564, 1.84847062897443005881527195053, 2.27441389998862881456962322526, 2.49529307446799026054557928504, 2.81681354362779972551415009849, 2.90150301558418022506540329796, 3.35970391375422249321562918347, 3.57063600973940416759050567102, 3.94593363704164899158115402591, 4.11512688321308378648967089916, 4.46965917503732683491478775816, 4.56935911595939851380123199415, 4.79876274307482899583782422644, 4.91331005623260117295370210204, 5.08739090476776641076545251497, 6.08019481742297483042924822901, 6.23078548815443055948851362510, 6.42239838291363031223512022359, 6.72170602454147341278508062384, 6.80543112905942585348483167052, 6.81267419863250867388740139920, 7.32592420002085790698621608933, 7.37889227873632408428905342381, 7.76807304728872810474613486327