Dirichlet series
| $L(s,\rho)$ = 1 | − 3-s − 7-s − 11-s + 21-s + 25-s + 27-s + 33-s + 37-s − 41-s − 47-s − 53-s + 2·67-s − 71-s − 73-s − 75-s + 77-s − 81-s − 83-s − 101-s + 2·107-s − 111-s + 123-s − 127-s + 2·137-s + 2·139-s + 141-s − 149-s + ⋯ |
Functional equation
\[\begin{aligned}
\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s)^{2} \, L(s,\rho)\cr
=\mathstrut & \, \Lambda(1-s)
\end{aligned}
\]
Invariants
| \( d \) | = | \(2\) |
| \( N \) | = | \(148\) = \(2^{2} \cdot 37\) |
| \( \varepsilon \) | = | $1$ |
| primitive | : | yes |
| self-dual | : | yes |
| Selberg data | = | $(2,\ 148,\ (0, 0:\ ),\ 1)$ |
Euler product
\[\begin{aligned}
L(s,\rho) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}
\end{aligned}\]
Particular Values
\[L(1/2,\rho) \approx 0.2355320219\]
\[L(1,\rho) \approx 0.5465728811\]