Dirichlet series
\[\begin{align}
L(s, E, \mathrm{sym}^3)=1\mathstrut& - 1.073 \ 5^{-s} + 0.053 \ 7^{-s} - 0.657 \ 11^{-s} + 0.938 \ 13^{-s} + 0.171 \ 17^{-s} + 1.062 \ 19^{-s} + 0.912 \ 25^{-s} - 0.691 \ 29^{-s}\cr
& - 0.057 \ 35^{-s} - 1.013 \ 37^{-s} + 0.594 \ 41^{-s} - 0.993 \ 43^{-s} + 0.002 \ 49^{-s} + 1.088 \ 53^{-s} + 0.706 \ 55^{-s} + 0.688 \ 59^{-s}\cr
& + 0.495 \ 61^{-s} - 1.007 \ 65^{-s} + 0.860 \ 67^{-s} + 1.058 \ 73^{-s} - 0.035 \ 77^{-s} + 2.233 \ 79^{-s} + 0.349 \ 83^{-s} - 0.183 \ 85^{-s}\cr
& + 0.300 \ 89^{-s} + 0.050 \ 91^{-s} - 1.140 \ 95^{-s} + \ \cdots
\end{align}\]
Functional equation
\[\begin{align}
\Lambda(s,E,\mathrm{sym}^{3})=\mathstrut & 7112448 ^{s/2} \Gamma_{\C}(s+1.5) \Gamma_{\C}(s+0.5) \cdot L(s, E, \mathrm{sym}^3)\cr
=\mathstrut & \Lambda(1-{s}, E,\mathrm{sym}^{3})
\end{align}
\]
Selberg data: $(4,7112448,(:1.5, 0.5), 1)$
Euler product
\[\begin{align} L(s,E, \mathrm{sym}^{3}) = & \prod_{p \nmid 1008 } \prod_{j=0}^{3} \left(1- \frac{\alpha_p^j\beta_p^{3-j}}{p^{s}}\right)^{-1} \\ & \times (1-7^{- s})^{-1}\end{align}\]
Imaginary part of the first few zeroes on the critical line
Particular Values
\[L(1/2, E, \mathrm{sym}^3) \approx 2.2905238705\]
\[L(1, E, \mathrm{sym}^3) \approx 0.9617090133\]
\[L(1, E, \mathrm{sym}^3) \approx 0.9617090133\]