L(s) = 1 | + 2·7-s − 16-s + 2·19-s − 2·31-s − 2·37-s + 2·49-s + 2·67-s + 2·73-s − 2·97-s + 2·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
L(s) = 1 | + 2·7-s − 16-s + 2·19-s − 2·31-s − 2·37-s + 2·49-s + 2·67-s + 2·73-s − 2·97-s + 2·109-s − 2·112-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.444585619\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444585619\) |
\(L(1)\) |
|
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
−9.771287775595117019428877122743, −9.492389179499621192347036867256, −8.936301586420276716704119228963, −8.680573982591645169164372269874, −8.292715549792970586562583242963, −7.84138029764644006451824924470, −7.46548761150637029561010310272, −7.13800899756430339627551879203, −6.84212553812636642622348832502, −6.17205468239825203767644146189, −5.44778760234932995012295545444, −5.31575483310630235484014456841, −4.98999663676339519289227836343, −4.58013366071585867010715589083, −3.77475559309054979235419571932, −3.64893983664295827796460757339, −2.85421897115998887987872250432, −2.05742189762647548783545866750, −1.81695550304147326141005723360, −1.07304974082192346023987081374,
1.07304974082192346023987081374, 1.81695550304147326141005723360, 2.05742189762647548783545866750, 2.85421897115998887987872250432, 3.64893983664295827796460757339, 3.77475559309054979235419571932, 4.58013366071585867010715589083, 4.98999663676339519289227836343, 5.31575483310630235484014456841, 5.44778760234932995012295545444, 6.17205468239825203767644146189, 6.84212553812636642622348832502, 7.13800899756430339627551879203, 7.46548761150637029561010310272, 7.84138029764644006451824924470, 8.292715549792970586562583242963, 8.680573982591645169164372269874, 8.936301586420276716704119228963, 9.492389179499621192347036867256, 9.771287775595117019428877122743