L(s) = 1 | + 3.47e6·4-s + 7.68e12·16-s + 5.90e13·19-s − 4.76e14·25-s − 1.64e16·31-s + 1.11e18·49-s + 9.60e18·61-s + 1.14e19·64-s + 2.05e20·76-s + 3.36e20·79-s − 1.65e21·100-s + 2.69e21·109-s − 1.48e22·121-s − 5.72e22·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.94e23·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | + 1.65·4-s + 1.74·16-s + 2.21·19-s − 25-s − 3.60·31-s + 2·49-s + 1.72·61-s + 1.23·64-s + 3.66·76-s + 3.99·79-s − 1.65·100-s + 1.09·109-s − 2·121-s − 5.97·124-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(6.087622547\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.087622547\) |
\(L(\frac{23}{2})\) |
|
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
−11.94990989411473051022532805445, −11.30328752233750272810599382058, −10.84880603976909862598727868849, −10.41308552491636412337055058240, −9.402209379582557831096666659781, −9.387493847404681578983761753043, −8.252103102674108818351787488563, −7.53133524794276029166181851641, −7.32237743766892152777113978686, −6.83907115468086695498348821913, −5.97189388587692062459713270149, −5.52454964221300856022833526358, −5.13314027105602053047432896454, −3.71959911526629136307525644024, −3.65174765729432043217799744428, −2.84700106178930892045948771056, −2.08811921860502155089513970832, −1.87499761754620523689147830506, −1.06680097203111146941919170743, −0.50738432469970482956404683227,
0.50738432469970482956404683227, 1.06680097203111146941919170743, 1.87499761754620523689147830506, 2.08811921860502155089513970832, 2.84700106178930892045948771056, 3.65174765729432043217799744428, 3.71959911526629136307525644024, 5.13314027105602053047432896454, 5.52454964221300856022833526358, 5.97189388587692062459713270149, 6.83907115468086695498348821913, 7.32237743766892152777113978686, 7.53133524794276029166181851641, 8.252103102674108818351787488563, 9.387493847404681578983761753043, 9.402209379582557831096666659781, 10.41308552491636412337055058240, 10.84880603976909862598727868849, 11.30328752233750272810599382058, 11.94990989411473051022532805445