L(s) = 1 | − 2·3-s + 3·9-s − 16-s − 4·27-s − 2·37-s + 2·48-s − 2·49-s + 5·81-s + 4·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 2·3-s + 3·9-s − 16-s − 4·27-s − 2·37-s + 2·48-s − 2·49-s + 5·81-s + 4·111-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 3·144-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2134578211\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2134578211\) |
\(L(1)\) |
|
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
−13.86662501688781585150006845396, −13.63389415332974561022405771071, −12.88032595670467214222071285001, −12.50712601288678636476174285911, −12.08169828841517647750547962500, −11.37051472087599925796301330751, −11.26336592177830283349310279878, −10.61941457244762634938626630511, −10.09400887276507614513350913030, −9.627286889041242972834937905847, −8.921363424316000325266869365009, −8.109234277758529601641032437684, −7.23298105014718581945665391039, −6.88701971246819184873514860915, −6.28258528956796865322833558727, −5.66648124971551170411397274205, −4.94230393511775675506662434397, −4.53540235685744608690914638694, −3.54574592690469057076498645767, −1.81269192173086389162999101018,
1.81269192173086389162999101018, 3.54574592690469057076498645767, 4.53540235685744608690914638694, 4.94230393511775675506662434397, 5.66648124971551170411397274205, 6.28258528956796865322833558727, 6.88701971246819184873514860915, 7.23298105014718581945665391039, 8.109234277758529601641032437684, 8.921363424316000325266869365009, 9.627286889041242972834937905847, 10.09400887276507614513350913030, 10.61941457244762634938626630511, 11.26336592177830283349310279878, 11.37051472087599925796301330751, 12.08169828841517647750547962500, 12.50712601288678636476174285911, 12.88032595670467214222071285001, 13.63389415332974561022405771071, 13.86662501688781585150006845396