L(s) = 1 | − 2·4-s − 5-s − 2·11-s + 5·13-s + 4·16-s + 17-s − 2·19-s + 2·20-s + 23-s + 25-s − 8·29-s − 31-s − 3·37-s − 7·41-s + 4·44-s − 47-s − 10·52-s + 8·53-s + 2·55-s + 7·61-s − 8·64-s − 5·65-s + 16·67-s − 2·68-s + 10·71-s + 8·73-s + 4·76-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s − 0.603·11-s + 1.38·13-s + 16-s + 0.242·17-s − 0.458·19-s + 0.447·20-s + 0.208·23-s + 1/5·25-s − 1.48·29-s − 0.179·31-s − 0.493·37-s − 1.09·41-s + 0.603·44-s − 0.145·47-s − 1.38·52-s + 1.09·53-s + 0.269·55-s + 0.896·61-s − 64-s − 0.620·65-s + 1.95·67-s − 0.242·68-s + 1.18·71-s + 0.936·73-s + 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37485 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37485 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.099376046\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.099376046\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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
−14.76559252307448, −14.38937126322712, −13.66260594877840, −13.35480880389955, −12.81644143634773, −12.47512835407167, −11.63551261392338, −11.16702649684653, −10.66797344896095, −10.06364510947698, −9.543914591508268, −8.856518134481282, −8.430077765276156, −8.090243033214240, −7.359623617687321, −6.741390550824430, −5.994284498566726, −5.336107680500828, −5.046016483794456, −3.961086367130372, −3.848004082564933, −3.155649505761919, −2.149671258779065, −1.269458573596549, −0.4272856811233123,
0.4272856811233123, 1.269458573596549, 2.149671258779065, 3.155649505761919, 3.848004082564933, 3.961086367130372, 5.046016483794456, 5.336107680500828, 5.994284498566726, 6.741390550824430, 7.359623617687321, 8.090243033214240, 8.430077765276156, 8.856518134481282, 9.543914591508268, 10.06364510947698, 10.66797344896095, 11.16702649684653, 11.63551261392338, 12.47512835407167, 12.81644143634773, 13.35480880389955, 13.66260594877840, 14.38937126322712, 14.76559252307448