L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 3·9-s + 2·11-s − 6·12-s + 5·16-s + 6·18-s + 4·22-s − 8·24-s − 25-s − 4·27-s + 6·32-s − 4·33-s + 9·36-s + 6·44-s + 2·47-s − 10·48-s − 2·50-s − 8·54-s + 2·59-s − 2·61-s + 7·64-s − 8·66-s − 4·71-s + ⋯ |
L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 4·6-s + 4·8-s + 3·9-s + 2·11-s − 6·12-s + 5·16-s + 6·18-s + 4·22-s − 8·24-s − 25-s − 4·27-s + 6·32-s − 4·33-s + 9·36-s + 6·44-s + 2·47-s − 10·48-s − 2·50-s − 8·54-s + 2·59-s − 2·61-s + 7·64-s − 8·66-s − 4·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.771445026\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.771445026\) |
\(L(1)\) |
|
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
−9.028101814651339983260523664253, −8.874727636736137797639156850024, −7.80316161093026387727967412457, −7.72288634045536267311636444078, −7.15544376123027019367274466455, −7.06955587634096588770728201470, −6.44048880554466320918856203709, −6.31600476447832181583345611750, −5.96042953052918766299839862796, −5.73329796274109485794391837452, −5.22703701737261334150039922251, −4.85999652486853502074512177184, −4.39730039811628908245755804644, −4.14683783861498269452519621091, −3.76335263207762939493006363177, −3.50485489618095978913996616713, −2.59351227088200858406988510676, −2.06006925360479751380487572977, −1.32696882187564700418419324080, −1.21723083943508021765365270515,
1.21723083943508021765365270515, 1.32696882187564700418419324080, 2.06006925360479751380487572977, 2.59351227088200858406988510676, 3.50485489618095978913996616713, 3.76335263207762939493006363177, 4.14683783861498269452519621091, 4.39730039811628908245755804644, 4.85999652486853502074512177184, 5.22703701737261334150039922251, 5.73329796274109485794391837452, 5.96042953052918766299839862796, 6.31600476447832181583345611750, 6.44048880554466320918856203709, 7.06955587634096588770728201470, 7.15544376123027019367274466455, 7.72288634045536267311636444078, 7.80316161093026387727967412457, 8.874727636736137797639156850024, 9.028101814651339983260523664253