L(s) = 1 | − 2·5-s − 16-s − 2·19-s + 3·25-s + 2·49-s + 2·80-s − 81-s + 4·95-s − 2·121-s − 4·125-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 + ⋯ |
L(s) = 1 | − 2·5-s − 16-s − 2·19-s + 3·25-s + 2·49-s + 2·80-s − 81-s + 4·95-s − 2·121-s − 4·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2489762189\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2489762189\) |
\(L(1)\) |
|
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
−14.61957954625341714164664019560, −14.13944646882214871532234905578, −13.27042701489436960383904648520, −12.93875799995861054376414921190, −12.22315444399224238023174054159, −11.97440438418630739376367126084, −11.35478032774755308995244767083, −10.72883796175334370510374909405, −10.64837931859429375334750285042, −9.566156585394324383794480118050, −8.691956235481506596317612803994, −8.588273787705263002746602261300, −7.88660575820453984407934978439, −7.14472277853861817875692871455, −6.82424315532089692950431228435, −5.92355506824304704240004536396, −4.75709052287312937184999225316, −4.29396961792893118367281974002, −3.66881192896887252833915429072, −2.51518215952811900062369427304,
2.51518215952811900062369427304, 3.66881192896887252833915429072, 4.29396961792893118367281974002, 4.75709052287312937184999225316, 5.92355506824304704240004536396, 6.82424315532089692950431228435, 7.14472277853861817875692871455, 7.88660575820453984407934978439, 8.588273787705263002746602261300, 8.691956235481506596317612803994, 9.566156585394324383794480118050, 10.64837931859429375334750285042, 10.72883796175334370510374909405, 11.35478032774755308995244767083, 11.97440438418630739376367126084, 12.22315444399224238023174054159, 12.93875799995861054376414921190, 13.27042701489436960383904648520, 14.13944646882214871532234905578, 14.61957954625341714164664019560