Curvy L with amssymb package

Comment \usepackage{mathptmx}.

Read page 5 in documentation of mathptmx. It says that the fonts for \mathcal symbols will be changed.

\documentclass[a4paper, 12pt]{book}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{mathtools}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{url}
\usepackage{framed}
\usepackage{setspace}
%\usepackage{mathptmx}
\usepackage{multirow}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{framed}

\begin{document}
\setlength{\parindent}{0cm}
\renewcommand{\baselinestretch}{0.5}

$\mathcal{L}$

\end{document}

If you adding the two rows

\usepackage{calrsfs}
\DeclareMathAlphabet{\pazocal}{OMS}{zplm}{m}{n}

it is possibile, in the same file .tex, to use the both curvy option. I have removed \usepackage{amsmath} with \usepackage{mathtools} because loading mathtools, you will have both packages loaded and you do not need to \usepackage{amsmath} in your preamble.

\documentclass[a4paper, 12pt]{book}
\usepackage{amssymb}
\usepackage{mathtools}
\usepackage[utf8]{inputenc}
\usepackage{geometry}
\usepackage{url}
\usepackage{framed}
\usepackage{setspace}
\usepackage{mathptmx}
\usepackage{multirow}
\usepackage{listings}
\usepackage{xcolor}
\usepackage{framed}

\usepackage{calrsfs}%%%%%%%%%%<------add
\DeclareMathAlphabet{\pazocal}{OMS}{zplm}{m}{n}%%%%%%%%%%<------add

\begin{document}
\setlength{\parindent}{0cm}
\renewcommand{\baselinestretch}{0.5}

$\mathcal{L}, \pazocal{L}$

\end{document}