L(s) = 1 | − 16-s − 81-s + 4·121-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 + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
L(s) = 1 | − 16-s − 81-s + 4·121-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 + 229-s + 233-s + 239-s + 241-s + 251-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5955645218\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5955645218\) |
\(L(1)\) |
|
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
−7.891029457169636612642187342351, −7.71542836175261784312940339271, −7.29711525702640386291335044664, −7.24158237298587336364111639163, −7.13116825587942048008052427899, −6.58737435379548800679241011965, −6.52609314592678570021388224505, −6.37184887812364498097616006368, −6.05759419280903329947891325220, −5.61989522060658386020253946963, −5.45707401313710585920192779646, −5.44616737155818135769791162795, −4.80417957660723987035738135581, −4.80159844471133660201479248304, −4.35853059717107534012039461718, −4.21470765862868866762739786483, −4.04886870152949275112666470514, −3.39863170052532323827461680497, −3.32480159013121887546173704780, −3.05179862368931192218855114284, −2.60587304832788196123186125509, −2.16481294294981664926714320973, −2.05537942047708543105388562806, −1.52492142328152317322830966531, −0.934443192434912603597736463747,
0.934443192434912603597736463747, 1.52492142328152317322830966531, 2.05537942047708543105388562806, 2.16481294294981664926714320973, 2.60587304832788196123186125509, 3.05179862368931192218855114284, 3.32480159013121887546173704780, 3.39863170052532323827461680497, 4.04886870152949275112666470514, 4.21470765862868866762739786483, 4.35853059717107534012039461718, 4.80159844471133660201479248304, 4.80417957660723987035738135581, 5.44616737155818135769791162795, 5.45707401313710585920192779646, 5.61989522060658386020253946963, 6.05759419280903329947891325220, 6.37184887812364498097616006368, 6.52609314592678570021388224505, 6.58737435379548800679241011965, 7.13116825587942048008052427899, 7.24158237298587336364111639163, 7.29711525702640386291335044664, 7.71542836175261784312940339271, 7.891029457169636612642187342351