Here, I use Bruno's \slantbox from Shear transform a "box" in conjunction with \mathscr.
EDITED to use John K's variant of \slantbox at Adjust custom made upright greek letters when used in subscripts in order to achieve better horizontal positioning within the \slantbox.
\documentclass{article}
\usepackage{mathrsfs}
\newsavebox\foobox
\newlength{\foodim}
\newcommand{\slantbox}[2][0]{\mbox{%
\sbox{\foobox}{#2}%
\foodim=#1\wd\foobox
\hskip \wd\foobox
\hskip -0.5\foodim
\pdfsave
\pdfsetmatrix{1 0 #1 1}%
\llap{\usebox{\foobox}}%
\pdfrestore
\hskip 0.5\foodim
}}
\def\Laplace{\slantbox[-.45]{$\mathscr{L}$}}
\begin{document}
\begin{equation}
\Laplace\{\sin(t)\} = \frac{1}{s^2 + 1}
\end{equation}
\end{document}

If one needs it to be able to scale down to \scriptscriptstyle, then this approach will work.
\documentclass{article}
\usepackage{mathrsfs,scalerel}
\newsavebox\foobox
\newlength{\foodim}
\newcommand{\slantbox}[2][0]{\mbox{%
\sbox{\foobox}{#2}%
\foodim=#1\wd\foobox
\hskip \wd\foobox
\hskip -0.5\foodim
\pdfsave
\pdfsetmatrix{1 0 #1 1}%
\llap{\usebox{\foobox}}%
\pdfrestore
\hskip 0.5\foodim
}}
\def\Laplace{\ThisStyle{\slantbox[-.45]{$\SavedStyle\mathscr{L}$}}}
\begin{document}
\begin{equation}
\Laplace\{\sin(t)\} = \frac{1}{s^2 + 1} \quad\scriptscriptstyle
\Laplace\{\sin(t)\} = \frac{1}{s^2 + 1}
\end{equation}
\end{document}

\mathscr{L}(script capital L)? – Przemysław Scherwentke Apr 21 '15 at 13:12