L(s) = 1 | − 4-s + 2·5-s + 16-s + 2·17-s − 2·20-s + 2·25-s + 2·29-s − 2·37-s − 2·41-s + 2·61-s − 64-s − 2·68-s − 2·73-s + 2·80-s − 81-s + 4·85-s + 2·97-s − 2·100-s − 2·109-s − 2·113-s − 2·116-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + ⋯ |
L(s) = 1 | − 4-s + 2·5-s + 16-s + 2·17-s − 2·20-s + 2·25-s + 2·29-s − 2·37-s − 2·41-s + 2·61-s − 64-s − 2·68-s − 2·73-s + 2·80-s − 81-s + 4·85-s + 2·97-s − 2·100-s − 2·109-s − 2·113-s − 2·116-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 4·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.844794739\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.844794739\) |
\(L(1)\) |
|
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
−8.988317510774996668541736067437, −8.697110020316634563007092859195, −8.218461130640259915870701265046, −8.202334424155764257956765858584, −7.50575887639937940949158821573, −6.97185243947968735825637832093, −6.81343652147486035026172025044, −6.21055863605209663476274700992, −5.94295530227994839951083086231, −5.40497688836295314121031880538, −5.37516842637644558425263657366, −4.90901771012926610417118480275, −4.60127189344037187120258141629, −3.71589776636210439291140395299, −3.63283968892761488034441673367, −2.82000327023422200456686663783, −2.76639974321406096970467182040, −1.70607229695763933127210759749, −1.58963277607082267870183528224, −0.894378655439629435642484357070,
0.894378655439629435642484357070, 1.58963277607082267870183528224, 1.70607229695763933127210759749, 2.76639974321406096970467182040, 2.82000327023422200456686663783, 3.63283968892761488034441673367, 3.71589776636210439291140395299, 4.60127189344037187120258141629, 4.90901771012926610417118480275, 5.37516842637644558425263657366, 5.40497688836295314121031880538, 5.94295530227994839951083086231, 6.21055863605209663476274700992, 6.81343652147486035026172025044, 6.97185243947968735825637832093, 7.50575887639937940949158821573, 8.202334424155764257956765858584, 8.218461130640259915870701265046, 8.697110020316634563007092859195, 8.988317510774996668541736067437