L(s) = 1 | + 0.517·2-s − 3-s − 1.73·4-s − 0.517·6-s − 1.93·8-s + 9-s + 3.46·11-s + 1.73·12-s + 4·13-s + 2.46·16-s + 4·17-s + 0.517·18-s − 5.27·19-s + 1.79·22-s − 3.48·23-s + 1.93·24-s + 2.07·26-s − 27-s − 4.92·29-s + 7.34·31-s + 5.13·32-s − 3.46·33-s + 2.07·34-s − 1.73·36-s − 10.5·37-s − 2.73·38-s − 4·39-s + ⋯ |
L(s) = 1 | + 0.366·2-s − 0.577·3-s − 0.866·4-s − 0.211·6-s − 0.683·8-s + 0.333·9-s + 1.04·11-s + 0.500·12-s + 1.10·13-s + 0.616·16-s + 0.970·17-s + 0.122·18-s − 1.21·19-s + 0.382·22-s − 0.726·23-s + 0.394·24-s + 0.406·26-s − 0.192·27-s − 0.915·29-s + 1.31·31-s + 0.908·32-s − 0.603·33-s + 0.355·34-s − 0.288·36-s − 1.73·37-s − 0.443·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.451946307\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.451946307\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.517T + 2T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 5.27T + 19T^{2} \) |
| 23 | \( 1 + 3.48T + 23T^{2} \) |
| 29 | \( 1 + 4.92T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 - 0.757T + 59T^{2} \) |
| 61 | \( 1 + 0.656T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 6.39T + 71T^{2} \) |
| 73 | \( 1 + 2.92T + 73T^{2} \) |
| 79 | \( 1 + 4.53T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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
−8.673766925345044562478214103191, −7.902450434068144026979063560622, −6.79316083441553513076678021937, −6.15814948404598506354642699595, −5.57136042257958697844227705808, −4.70414068967458003378249543241, −3.88506587307919313930732029910, −3.45882197710195971320529396808, −1.82008814499021281810217543221, −0.71414800560599163514864004153,
0.71414800560599163514864004153, 1.82008814499021281810217543221, 3.45882197710195971320529396808, 3.88506587307919313930732029910, 4.70414068967458003378249543241, 5.57136042257958697844227705808, 6.15814948404598506354642699595, 6.79316083441553513076678021937, 7.902450434068144026979063560622, 8.673766925345044562478214103191