L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 2·25-s − 6·32-s + 2·41-s + 4·50-s − 4·61-s + 7·64-s − 81-s − 4·82-s − 6·100-s + 8·122-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 6·164-s + 167-s + 2·169-s + ⋯ |
L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·16-s − 2·25-s − 6·32-s + 2·41-s + 4·50-s − 4·61-s + 7·64-s − 81-s − 4·82-s − 6·100-s + 8·122-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·162-s + 163-s + 6·164-s + 167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2002762706\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2002762706\) |
\(L(1)\) |
|
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
−13.18763867837712168221678951560, −12.51833864361520564569950340750, −12.16056558853819033271782428387, −11.59109040205698406505356516171, −11.25219882261673457397142875486, −10.64515854163204002054397390352, −10.30712458699540265229593530298, −9.694866627355286219347224837793, −9.187105519113440642461821141702, −9.024266535468335828680798604443, −7.958130159799043255838239927925, −7.955392826552363289980170655329, −7.33632915986505954677530283474, −6.69921581546472461321751958786, −5.85609162058118679786749317933, −5.83779551634800273249445863948, −4.40272053803656526702806000032, −3.38262324183870129332137983046, −2.53413337667629057470742593665, −1.61066445217681730691510386296,
1.61066445217681730691510386296, 2.53413337667629057470742593665, 3.38262324183870129332137983046, 4.40272053803656526702806000032, 5.83779551634800273249445863948, 5.85609162058118679786749317933, 6.69921581546472461321751958786, 7.33632915986505954677530283474, 7.955392826552363289980170655329, 7.958130159799043255838239927925, 9.024266535468335828680798604443, 9.187105519113440642461821141702, 9.694866627355286219347224837793, 10.30712458699540265229593530298, 10.64515854163204002054397390352, 11.25219882261673457397142875486, 11.59109040205698406505356516171, 12.16056558853819033271782428387, 12.51833864361520564569950340750, 13.18763867837712168221678951560