L(s) = 1 | + 2·2-s + 2·4-s + 2·8-s − 9-s + 3·16-s − 2·18-s + 2·25-s − 2·31-s + 4·32-s − 2·36-s − 2·41-s + 2·47-s + 2·49-s + 4·50-s − 4·62-s + 4·64-s − 2·72-s − 2·73-s + 81-s − 4·82-s + 4·94-s + 4·98-s + 4·100-s − 2·101-s − 4·124-s + 127-s + 4·128-s + ⋯ |
L(s) = 1 | + 2·2-s + 2·4-s + 2·8-s − 9-s + 3·16-s − 2·18-s + 2·25-s − 2·31-s + 4·32-s − 2·36-s − 2·41-s + 2·47-s + 2·49-s + 4·50-s − 4·62-s + 4·64-s − 2·72-s − 2·73-s + 81-s − 4·82-s + 4·94-s + 4·98-s + 4·100-s − 2·101-s − 4·124-s + 127-s + 4·128-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.739461862\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.739461862\) |
\(L(1)\) |
|
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
−9.448714402438412153078856361613, −9.013583199878180017521722825469, −8.818386907003194553262343213001, −8.311054210908338613961222226780, −7.909930291327359386067163393486, −7.38519636203980971489244978722, −7.01173845027374617510831424490, −6.82301674379719791647076645185, −6.08930402049363758613191143349, −5.77487939191565230315492397537, −5.46915395071510417297680507734, −5.06090099359527545749738494172, −4.84606035679535512913514622259, −4.14873997327109448197734214665, −3.85620080643315639556510783474, −3.48871735581238695571082916185, −2.76570478491314705304799496869, −2.71534346064884293610064227614, −1.80863970995744292176853033784, −1.11946198236062556720060919945,
1.11946198236062556720060919945, 1.80863970995744292176853033784, 2.71534346064884293610064227614, 2.76570478491314705304799496869, 3.48871735581238695571082916185, 3.85620080643315639556510783474, 4.14873997327109448197734214665, 4.84606035679535512913514622259, 5.06090099359527545749738494172, 5.46915395071510417297680507734, 5.77487939191565230315492397537, 6.08930402049363758613191143349, 6.82301674379719791647076645185, 7.01173845027374617510831424490, 7.38519636203980971489244978722, 7.909930291327359386067163393486, 8.311054210908338613961222226780, 8.818386907003194553262343213001, 9.013583199878180017521722825469, 9.448714402438412153078856361613