Scenario 1: Conversion Between Pearson and Spearman Correlation Coefficients
Pearson Correlation Coefficient   $r$
    Spearman Correlation Coefficient  $r_s$

Convert Pearson Correlation Coefficient $r$ to Spearman Correlation Coefficient $r_s$

$$r_s=\frac{6}{\pi}*sin^{-1}(\frac{r}{2})$$

Convert Spearman Correlation Coefficient $r_s$ to Pearson Correlation Coefficient $r$

$$r=2*sin(r_s*\frac{\pi}{6})$$
Scenario 2: Conversion Between Pearson and Kendall Correlation Coefficients
Pearson Correlation Coefficient   $r$
    Kendall Correlation Coefficient   $\tau$

Convert Pearson Correlation Coefficient $r$ to KendallCorrelation Coefficient $\tau$

$$\tau=\frac{2}{\pi}*sin^{-1}(r)$$

Convert KendallCorrelation Coefficient $\tau$ to PearsonCorrelation Coefficient $r$

$$r=sin(\tau*\frac{\pi}{2})$$

Reference:

[1] Rupinski, M. T., & Dunlap, W. P. (1996). Approximating Pearson Product-Moment Correlations from Kendall’s Tau and Spearman’s Rho. Educational and Psychological Measurement, 56(3), 419–429.
[2] Chen, Lihua et al. “The correlation between apparent diffusion coefficient and tumor cellularity in patients: a meta-analysis.” PloS one vol. 8,11 e79008. 11 Nov. 2013, doi:10.1371/journal.pone.0079008