L(s) = 1 | − 2·11-s − 16-s − 2·19-s − 25-s + 2·31-s + 2·41-s + 2·49-s + 2·61-s − 2·79-s − 81-s − 2·89-s − 2·101-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 2·176-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 2·11-s − 16-s − 2·19-s − 25-s + 2·31-s + 2·41-s + 2·49-s + 2·61-s − 2·79-s − 81-s − 2·89-s − 2·101-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 2·176-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3755910542\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3755910542\) |
\(L(1)\) |
|
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
−13.42152706905986775804239387810, −13.19007158956853907013601278393, −12.60653029703028197500961535334, −12.24017337137894231430027189121, −11.40313183987310958416432208044, −11.10773304512960009381429154755, −10.47759928528073937459323626242, −10.15441551610510815076338042987, −9.608335148010529482036674461967, −8.766585948101750623257275695389, −8.365995397532859272406981295165, −7.929057424431366116466128070148, −7.21612790594965992491197968021, −6.66425983048885149547281387292, −5.85627077927153378918702552554, −5.46047728948515655924322566353, −4.37461038896300874501784182487, −4.23963785046997912056196187044, −2.66446834598452832199469494415, −2.37441829380017958106060039538,
2.37441829380017958106060039538, 2.66446834598452832199469494415, 4.23963785046997912056196187044, 4.37461038896300874501784182487, 5.46047728948515655924322566353, 5.85627077927153378918702552554, 6.66425983048885149547281387292, 7.21612790594965992491197968021, 7.929057424431366116466128070148, 8.365995397532859272406981295165, 8.766585948101750623257275695389, 9.608335148010529482036674461967, 10.15441551610510815076338042987, 10.47759928528073937459323626242, 11.10773304512960009381429154755, 11.40313183987310958416432208044, 12.24017337137894231430027189121, 12.60653029703028197500961535334, 13.19007158956853907013601278393, 13.42152706905986775804239387810