L(s) = 1 | − 3.89e6·4-s − 3.48e9·9-s + 2.91e10·11-s + 1.07e13·16-s + 5.84e13·19-s − 4.80e15·29-s + 4.47e15·31-s + 1.35e16·36-s − 2.06e17·41-s − 1.13e17·44-s + 9.85e17·49-s − 1.10e19·59-s − 1.43e19·61-s − 2.47e19·64-s + 5.29e19·71-s − 2.27e20·76-s + 3.37e19·79-s + 1.21e19·81-s + 6.25e20·89-s − 1.01e20·99-s + 2.89e20·101-s − 4.49e20·109-s + 1.86e22·116-s − 1.41e22·121-s − 1.74e22·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1.85·4-s − 1/3·9-s + 0.339·11-s + 2.44·16-s + 2.18·19-s − 2.11·29-s + 0.981·31-s + 0.618·36-s − 2.40·41-s − 0.629·44-s + 1.76·49-s − 2.81·59-s − 2.57·61-s − 2.68·64-s + 1.92·71-s − 4.05·76-s + 0.401·79-s + 1/9·81-s + 2.12·89-s − 0.113·99-s + 0.260·101-s − 0.181·109-s + 3.93·116-s − 1.91·121-s − 1.82·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(1.230730571\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230730571\) |
\(L(\frac{23}{2})\) |
|
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
−10.70395271595365656400159869689, −10.31790327359193048529385548741, −9.540105257839947541989109318210, −9.213803361351557768225144693957, −9.136884268012323542494651105478, −8.203183620132552849886121851833, −7.85707229564663456303443403208, −7.36437350569723422554403866589, −6.55895443587506111554054254622, −5.67961360659597809460663018095, −5.55083429288188830203847058749, −4.78787285958715234041296435553, −4.57260614924339785961083986405, −3.59590640501014609903974231089, −3.49597279815061322255561467576, −2.90913331024900715429975787473, −1.83237557819244857294713399103, −1.35881730184474954745301644058, −0.72897532887133491439856541628, −0.29829578101666617833321259049,
0.29829578101666617833321259049, 0.72897532887133491439856541628, 1.35881730184474954745301644058, 1.83237557819244857294713399103, 2.90913331024900715429975787473, 3.49597279815061322255561467576, 3.59590640501014609903974231089, 4.57260614924339785961083986405, 4.78787285958715234041296435553, 5.55083429288188830203847058749, 5.67961360659597809460663018095, 6.55895443587506111554054254622, 7.36437350569723422554403866589, 7.85707229564663456303443403208, 8.203183620132552849886121851833, 9.136884268012323542494651105478, 9.213803361351557768225144693957, 9.540105257839947541989109318210, 10.31790327359193048529385548741, 10.70395271595365656400159869689