L(s) = 1 | + 2·3-s − 2·4-s − 2·7-s + 2·9-s − 4·12-s + 3·16-s + 2·17-s − 4·21-s + 2·27-s + 4·28-s − 4·36-s − 2·37-s − 2·47-s + 6·48-s + 2·49-s + 4·51-s + 2·61-s − 4·63-s − 4·64-s − 4·68-s + 2·71-s + 2·79-s + 3·81-s + 8·84-s + 4·89-s − 2·97-s − 4·108-s + ⋯ |
L(s) = 1 | + 2·3-s − 2·4-s − 2·7-s + 2·9-s − 4·12-s + 3·16-s + 2·17-s − 4·21-s + 2·27-s + 4·28-s − 4·36-s − 2·37-s − 2·47-s + 6·48-s + 2·49-s + 4·51-s + 2·61-s − 4·63-s − 4·64-s − 4·68-s + 2·71-s + 2·79-s + 3·81-s + 8·84-s + 4·89-s − 2·97-s − 4·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 638401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 638401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9106667539\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9106667539\) |
\(L(1)\) |
|
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
−10.30813916166489070980551840416, −10.02162990233791658777688878230, −9.594040985757889731963605525407, −9.385843929804427785077256885783, −9.149004211195580521395692797172, −8.517647138644809170150742433872, −8.386354777601009821054263059587, −7.80485911257533043226855360494, −7.67952333185749838503850381614, −6.70849618064936116204528590558, −6.56318741009394828702748829611, −5.74635104893792175643103972437, −5.08622314534893812886407372604, −5.00742648746650945834870031272, −3.86665528610201218482511266896, −3.71067770523075155581245407326, −3.20984728687798741128797199671, −3.19389671564216599440055642828, −2.16326964126628773584340625955, −0.974320373461786274110774279262,
0.974320373461786274110774279262, 2.16326964126628773584340625955, 3.19389671564216599440055642828, 3.20984728687798741128797199671, 3.71067770523075155581245407326, 3.86665528610201218482511266896, 5.00742648746650945834870031272, 5.08622314534893812886407372604, 5.74635104893792175643103972437, 6.56318741009394828702748829611, 6.70849618064936116204528590558, 7.67952333185749838503850381614, 7.80485911257533043226855360494, 8.386354777601009821054263059587, 8.517647138644809170150742433872, 9.149004211195580521395692797172, 9.385843929804427785077256885783, 9.594040985757889731963605525407, 10.02162990233791658777688878230, 10.30813916166489070980551840416