L(s) = 1 | − 4-s − 2·11-s + 16-s − 2·29-s − 2·37-s + 2·43-s + 2·44-s − 49-s + 2·53-s − 64-s + 2·67-s − 81-s + 2·107-s + 2·109-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·172-s + ⋯ |
L(s) = 1 | − 4-s − 2·11-s + 16-s − 2·29-s − 2·37-s + 2·43-s + 2·44-s − 49-s + 2·53-s − 64-s + 2·67-s − 81-s + 2·107-s + 2·109-s + 2·116-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·172-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2824099001\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2824099001\) |
\(L(1)\) |
|
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
−14.30069270320800848660558005175, −13.45286471755867347856734524811, −13.13005281311943114418937395804, −12.75795018895155456044755509988, −12.31230408122088277357194352773, −11.49301810311495345341512613038, −10.90122155092304599712070645171, −10.42149304749246711383091061266, −9.968337038803622567918569441583, −9.372527297341524736564459728269, −8.712784745011879189339577700650, −8.319056652516069500791959709910, −7.48250950853313495465445459749, −7.34148884483076194341677592755, −6.09862088320331059470531847771, −5.28799620715987171801146708145, −5.22631067709461204170447629166, −4.13114150743211819775629176162, −3.39688143770569238837513800590, −2.27803974715281569194064287612,
2.27803974715281569194064287612, 3.39688143770569238837513800590, 4.13114150743211819775629176162, 5.22631067709461204170447629166, 5.28799620715987171801146708145, 6.09862088320331059470531847771, 7.34148884483076194341677592755, 7.48250950853313495465445459749, 8.319056652516069500791959709910, 8.712784745011879189339577700650, 9.372527297341524736564459728269, 9.968337038803622567918569441583, 10.42149304749246711383091061266, 10.90122155092304599712070645171, 11.49301810311495345341512613038, 12.31230408122088277357194352773, 12.75795018895155456044755509988, 13.13005281311943114418937395804, 13.45286471755867347856734524811, 14.30069270320800848660558005175