L(s) = 1 | − 2·5-s + 2·17-s − 25-s − 4·29-s + 2·37-s − 8·41-s − 7·49-s + 4·53-s + 12·61-s − 16·73-s − 4·85-s + 16·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.485·17-s − 1/5·25-s − 0.742·29-s + 0.328·37-s − 1.24·41-s − 49-s + 0.549·53-s + 1.53·61-s − 1.87·73-s − 0.433·85-s + 1.69·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7437493018\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7437493018\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.25546127955879, −11.97289093990341, −11.66177227028994, −11.12539355375774, −10.78440326746831, −10.09488518580366, −9.875127868511230, −9.271224357569998, −8.767635488781683, −8.289941253445130, −7.918104594170816, −7.449403840639517, −7.053821085287925, −6.494306245657989, −5.999356486223569, −5.380234401449326, −5.035498087663906, −4.385413539977461, −3.862728432720932, −3.548411277222912, −2.950669994195244, −2.335866259353213, −1.660425603093515, −1.070416257969322, −0.2386474050756605,
0.2386474050756605, 1.070416257969322, 1.660425603093515, 2.335866259353213, 2.950669994195244, 3.548411277222912, 3.862728432720932, 4.385413539977461, 5.035498087663906, 5.380234401449326, 5.999356486223569, 6.494306245657989, 7.053821085287925, 7.449403840639517, 7.918104594170816, 8.289941253445130, 8.767635488781683, 9.271224357569998, 9.875127868511230, 10.09488518580366, 10.78440326746831, 11.12539355375774, 11.66177227028994, 11.97289093990341, 12.25546127955879