L(s) = 1 | − 16-s − 4·23-s − 4·43-s − 4·53-s + 4·67-s + 4·107-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
L(s) = 1 | − 16-s − 4·23-s − 4·43-s − 4·53-s + 4·67-s + 4·107-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2555778200\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2555778200\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.545982489762076035360758359071, −8.823579721935180019121213818173, −8.469677166565118245229208814431, −8.434536957138598386964682300441, −8.331101550051849999487007881591, −7.81307626176211525269068538871, −7.69871991938462507386883068052, −7.58349170248394100224433670765, −6.88124373243444318155137645877, −6.80257890134922510809578542478, −6.42454372208400273938871110723, −6.29893581866913974948321944843, −6.10004080784686032476692554731, −5.63213365356674983441617051222, −5.34425991198805266625831792784, −4.84629761488409957491405872450, −4.81090955923048536674171304725, −4.38902995657153048539593973232, −4.05555792793186239881001508243, −3.55072378156165014876097734764, −3.42045103322412422864168502433, −3.05489877579825797630944806539, −2.21337394587585197564071831515, −1.87327003141636038599724248583, −1.86659789505119545782081617086,
1.86659789505119545782081617086, 1.87327003141636038599724248583, 2.21337394587585197564071831515, 3.05489877579825797630944806539, 3.42045103322412422864168502433, 3.55072378156165014876097734764, 4.05555792793186239881001508243, 4.38902995657153048539593973232, 4.81090955923048536674171304725, 4.84629761488409957491405872450, 5.34425991198805266625831792784, 5.63213365356674983441617051222, 6.10004080784686032476692554731, 6.29893581866913974948321944843, 6.42454372208400273938871110723, 6.80257890134922510809578542478, 6.88124373243444318155137645877, 7.58349170248394100224433670765, 7.69871991938462507386883068052, 7.81307626176211525269068538871, 8.331101550051849999487007881591, 8.434536957138598386964682300441, 8.469677166565118245229208814431, 8.823579721935180019121213818173, 9.545982489762076035360758359071