L(s) = 1 | + 1.09e12·4-s − 5.45e16·7-s − 4.05e18·9-s + 9.06e23·16-s − 3.63e27·25-s − 6.00e28·28-s − 4.45e30·36-s + 1.03e30·37-s + 1.24e32·43-s + 2.07e33·49-s + 2.21e35·63-s + 6.64e35·64-s + 7.22e35·67-s + 2.18e37·79-s + 1.64e37·81-s − 3.99e39·100-s − 1.21e40·109-s − 4.95e40·112-s + 8.22e40·121-s + 127-s + 131-s + 137-s + 139-s − 3.67e42·144-s + 1.13e42·148-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 2·4-s − 1.81·7-s − 9-s + 3·16-s − 2·25-s − 3.62·28-s − 2·36-s + 0.271·37-s + 1.75·43-s + 2.27·49-s + 1.81·63-s + 4·64-s + 1.78·67-s + 2.16·79-s + 81-s − 4·100-s − 2.25·109-s − 5.43·112-s + 2·121-s − 3·144-s + 0.542·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+39/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(20)\) |
\(\approx\) |
\(4.964579402\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.964579402\) |
\(L(\frac{41}{2})\) |
|
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
−11.08032216004088183742248738201, −11.06141554185866657944236902544, −10.18941107291136916475295904162, −9.724489876390417096211203571782, −9.248608114800294806487062099725, −8.293243868561784737823190532676, −7.75415805842596426854711322157, −7.24996454380325352572065411446, −6.54515711977350486475237043970, −6.35020690633978264816665195188, −5.61255075580973117695747334206, −5.60082413014624575458279490187, −4.15090282613603660968882864534, −3.51476135388203516136593097706, −3.21990280785127080963323779897, −2.55958447963267854874124540278, −2.26873826374979445380712122208, −1.72030802078574462773256295154, −0.68474147091730658447151654117, −0.52858109233238948700668611307,
0.52858109233238948700668611307, 0.68474147091730658447151654117, 1.72030802078574462773256295154, 2.26873826374979445380712122208, 2.55958447963267854874124540278, 3.21990280785127080963323779897, 3.51476135388203516136593097706, 4.15090282613603660968882864534, 5.60082413014624575458279490187, 5.61255075580973117695747334206, 6.35020690633978264816665195188, 6.54515711977350486475237043970, 7.24996454380325352572065411446, 7.75415805842596426854711322157, 8.293243868561784737823190532676, 9.248608114800294806487062099725, 9.724489876390417096211203571782, 10.18941107291136916475295904162, 11.06141554185866657944236902544, 11.08032216004088183742248738201