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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1773206590\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1773206590\) |
\(L(1)\) |
|
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
−15.59461919004984204952321323569, −14.84625762244638749011823645362, −14.21237496700354657276706814216, −13.73708928432188607873060131214, −13.15884627493339383429814612152, −12.36784286101715037183258870678, −12.21051665090741868031115105838, −11.54553781733229543264173850657, −10.68372419403733204059333291826, −10.67475820275141773886786578035, −9.401820822599483704682833492791, −8.986147440681671677679816455417, −8.208046929617629467128032615792, −8.063910698916515610767798864707, −7.09193503008472694559106097842, −6.52495137986341477590907454457, −5.27019738107696468255483049268, −4.29878395807902601847606510864, −4.22564061217201658866752193569, −3.03544157255568814852703229915,
3.03544157255568814852703229915, 4.22564061217201658866752193569, 4.29878395807902601847606510864, 5.27019738107696468255483049268, 6.52495137986341477590907454457, 7.09193503008472694559106097842, 8.063910698916515610767798864707, 8.208046929617629467128032615792, 8.986147440681671677679816455417, 9.401820822599483704682833492791, 10.67475820275141773886786578035, 10.68372419403733204059333291826, 11.54553781733229543264173850657, 12.21051665090741868031115105838, 12.36784286101715037183258870678, 13.15884627493339383429814612152, 13.73708928432188607873060131214, 14.21237496700354657276706814216, 14.84625762244638749011823645362, 15.59461919004984204952321323569