L(s) = 1 | − 1.31e5·4-s − 7.04e5·7-s + 9.70e9·13-s + 1.71e10·16-s − 9.09e10·19-s + 7.20e12·25-s + 9.23e10·28-s − 3.78e13·31-s − 3.33e14·37-s − 1.46e15·43-s − 3.25e15·49-s − 1.27e15·52-s − 1.38e16·61-s − 2.25e15·64-s + 2.45e16·67-s + 1.90e17·73-s + 1.19e16·76-s + 4.50e17·79-s − 6.83e15·91-s − 8.51e17·97-s − 9.44e17·100-s − 1.51e18·103-s + 1.05e18·109-s − 1.21e16·112-s + 1.02e19·121-s + 4.96e18·124-s + 127-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 0.0174·7-s + 0.915·13-s + 1/4·16-s − 0.281·19-s + 1.88·25-s + 0.00872·28-s − 1.43·31-s − 2.56·37-s − 2.90·43-s − 1.99·49-s − 0.457·52-s − 1.18·61-s − 1/8·64-s + 0.903·67-s + 3.23·73-s + 0.140·76-s + 3.75·79-s − 0.0159·91-s − 1.11·97-s − 0.944·100-s − 1.15·103-s + 0.483·109-s − 0.00436·112-s + 1.84·121-s + 0.716·124-s + 0.00491·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+9)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(1.345906913\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.345906913\) |
\(L(10)\) |
|
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
−15.03175397167232432771708484754, −13.73149719427354199688736963149, −13.73126119388606724368470108897, −12.55205409793819038241942697098, −12.43786047838454099405547073324, −11.17061636116092340925188640707, −10.83856949771865856809733701952, −9.990350357854369899987429494137, −9.188741971764477595774743185259, −8.566585188982975339605036675028, −8.031933836453856949101132045543, −6.82819293845238976885637730989, −6.47608283722096441712418219654, −5.13684326771282464154784097889, −4.98101607698471126849341797032, −3.54096213677051747611122838602, −3.41517275044166097817268267507, −2.00088030014253334547811176577, −1.33294331744059374531782457721, −0.35816941344330883538541614417,
0.35816941344330883538541614417, 1.33294331744059374531782457721, 2.00088030014253334547811176577, 3.41517275044166097817268267507, 3.54096213677051747611122838602, 4.98101607698471126849341797032, 5.13684326771282464154784097889, 6.47608283722096441712418219654, 6.82819293845238976885637730989, 8.031933836453856949101132045543, 8.566585188982975339605036675028, 9.188741971764477595774743185259, 9.990350357854369899987429494137, 10.83856949771865856809733701952, 11.17061636116092340925188640707, 12.43786047838454099405547073324, 12.55205409793819038241942697098, 13.73126119388606724368470108897, 13.73149719427354199688736963149, 15.03175397167232432771708484754