L(s) = 1 | + 1.71e10·4-s − 1.58e12·7-s − 5.55e15·9-s + 2.21e20·16-s − 2.32e23·25-s − 2.73e22·28-s − 9.55e25·36-s + 1.24e26·37-s − 1.19e27·43-s − 7.72e27·49-s + 8.83e27·63-s + 2.53e30·64-s − 3.93e30·67-s − 7.64e31·79-s + 3.09e31·81-s − 3.99e33·100-s + 3.99e32·109-s − 3.51e32·112-s + 4.64e34·121-s + 127-s + 131-s + 137-s + 139-s − 1.23e36·144-s + 2.13e36·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 2·4-s − 0.0180·7-s − 9-s + 3·16-s − 2·25-s − 0.0361·28-s − 2·36-s + 1.65·37-s − 1.33·43-s − 0.999·49-s + 0.0180·63-s + 4·64-s − 2.91·67-s − 3.73·79-s + 81-s − 4·100-s + 0.0963·109-s − 0.0542·112-s + 2·121-s − 3·144-s + 3.30·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+33/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(17)\) |
\(\approx\) |
\(1.240683769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.240683769\) |
\(L(\frac{35}{2})\) |
|
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
−11.75597968710494171526101057403, −11.36319604855647641634955305864, −11.08355023186861475243998315062, −10.03463981133916323379650735865, −10.00791387526172531542949872125, −8.882101089519224247129425864881, −8.184144687933416395144613324777, −7.64329256812074160581944276351, −7.26992270578374018833452595133, −6.39072102657627412142429426524, −6.01991648491591392154873020392, −5.70823230109624286063943868437, −4.80337174425288640686307785829, −3.85022615922826432676959765542, −3.28788307723655027109184455758, −2.64432656460013503773393470703, −2.40186040588091505622487788367, −1.47879186185359150905467252630, −1.37191212566506567442136201734, −0.18965131431482006915860453698,
0.18965131431482006915860453698, 1.37191212566506567442136201734, 1.47879186185359150905467252630, 2.40186040588091505622487788367, 2.64432656460013503773393470703, 3.28788307723655027109184455758, 3.85022615922826432676959765542, 4.80337174425288640686307785829, 5.70823230109624286063943868437, 6.01991648491591392154873020392, 6.39072102657627412142429426524, 7.26992270578374018833452595133, 7.64329256812074160581944276351, 8.184144687933416395144613324777, 8.882101089519224247129425864881, 10.00791387526172531542949872125, 10.03463981133916323379650735865, 11.08355023186861475243998315062, 11.36319604855647641634955305864, 11.75597968710494171526101057403