Dirichlet series
| $L(s,\rho)$ = 1 | + 2-s + 4-s + 8-s − 9-s + 16-s − 17-s − 18-s − 25-s + 32-s − 34-s − 36-s + 2·47-s + 49-s − 50-s + 64-s − 68-s − 72-s + 81-s + 2·89-s + 2·94-s + 98-s − 100-s − 2·103-s − 121-s − 2·127-s + 128-s − 136-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\R}(s+1)^{2} \, L(s,\rho)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(136\) = \(2^{3} \cdot 17\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 136,\ (1, 1:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,\rho) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Particular Values
\[L(1/2,\rho) \approx 1.946451495\]
\[L(1,\rho) \approx 1.692623190\]