L(s) = 1 | − 4.30e7·3-s − 4.29e9·4-s + 6.51e13·7-s + 1.84e17·12-s + 2.44e17·13-s + 7.82e18·19-s − 2.80e21·21-s − 2.32e22·25-s + 7.97e22·27-s − 2.79e23·28-s + 1.41e24·31-s + 2.15e25·37-s − 1.05e25·39-s + 5.45e26·43-s + 3.14e27·49-s − 1.04e27·52-s − 3.36e26·57-s + 3.95e28·61-s + 7.92e28·64-s − 5.31e27·67-s − 7.67e29·73-s + 1.00e30·75-s − 3.35e28·76-s − 2.02e30·79-s − 3.43e30·81-s + 1.20e31·84-s + 1.59e31·91-s + ⋯ |
L(s) = 1 | − 3-s − 4-s + 1.96·7-s + 12-s + 0.367·13-s + 0.0271·19-s − 1.96·21-s − 25-s + 27-s − 1.96·28-s + 1.94·31-s + 1.74·37-s − 0.367·39-s + 3.99·43-s + 2.84·49-s − 0.367·52-s − 0.0271·57-s + 1.07·61-s + 64-s − 0.0322·67-s − 1.18·73-s + 75-s − 0.0271·76-s − 0.880·79-s − 81-s + 1.96·84-s + 0.719·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+16)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(2.628842828\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.628842828\) |
\(L(17)\) |
|
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
−11.93123616594753840638631511863, −11.25977208872065284165053789426, −11.22404101111873193513511153439, −10.38839585772668743983918857480, −9.761494789861261556376771884386, −8.795514513826156545547640106280, −8.689233516055067288426319978044, −7.67351451001015672166724938868, −7.63279756437076862540639556802, −6.33967424979526184532554768217, −5.88955765549239595922325814973, −5.34329495517252849127077522964, −4.70701965937614222843312961314, −4.34519796733871313121492236275, −3.95482219675028789688841558724, −2.63899092748524125086599709242, −2.27803599298492874428520824623, −1.25463829816007482329923375189, −0.879249101903780611763628975886, −0.48501811786860854016447992974,
0.48501811786860854016447992974, 0.879249101903780611763628975886, 1.25463829816007482329923375189, 2.27803599298492874428520824623, 2.63899092748524125086599709242, 3.95482219675028789688841558724, 4.34519796733871313121492236275, 4.70701965937614222843312961314, 5.34329495517252849127077522964, 5.88955765549239595922325814973, 6.33967424979526184532554768217, 7.63279756437076862540639556802, 7.67351451001015672166724938868, 8.689233516055067288426319978044, 8.795514513826156545547640106280, 9.761494789861261556376771884386, 10.38839585772668743983918857480, 11.22404101111873193513511153439, 11.25977208872065284165053789426, 11.93123616594753840638631511863