I am reading Harald Cramér's 1919 paper. On page 111, he wrote that by plugging the functional equation for $\zeta(s)$ into ($C=\varepsilon,B=i\varepsilon,A=+i\infty$): $$ I=z\int_{CBA}e^{zs}\log\zeta(s)\mathrm ds, $$ there is $$ I=z\int_{CBA}e^{sz}\left[s\log2\pi-\log\pi+\log\sin\left(\pi s\over2\right)+\log\Gamma(1-s)+\log\zeta(1-s)\right]\mathrm ds. $$ I am able to follow this argument, but the my concern is on the next step. On the paragraph between (8) and (8a) of his paper, Cramér argues that one needs to correct the above expression into $$ I=z\int_{CBA}e^{zs}\left[s\log2\pi-\log\pi+\log\sin\left(\pi s\over2\right)+\log\Gamma(1-s)+\log\zeta(1-s)\color{red}{-2\pi i}\right]\mathrm ds. $$
in order to let the integrand "vary continuously." As I am not familiar with German, I am not able to follow his reasoning. I wonder whether there are simple explanations on why the red term needs to appear.