L(s) = 1 | − 2.75·2-s + 9·3-s − 24.4·4-s + 5.39·5-s − 24.8·6-s − 153.·7-s + 155.·8-s + 81·9-s − 14.8·10-s + 761.·11-s − 219.·12-s + 217.·13-s + 423.·14-s + 48.5·15-s + 352.·16-s − 1.05e3·17-s − 223.·18-s − 2.36e3·19-s − 131.·20-s − 1.38e3·21-s − 2.09e3·22-s + 3.57e3·23-s + 1.39e3·24-s − 3.09e3·25-s − 600.·26-s + 729·27-s + 3.74e3·28-s + ⋯ |
L(s) = 1 | − 0.487·2-s + 0.577·3-s − 0.762·4-s + 0.0965·5-s − 0.281·6-s − 1.18·7-s + 0.858·8-s + 0.333·9-s − 0.0470·10-s + 1.89·11-s − 0.440·12-s + 0.357·13-s + 0.576·14-s + 0.0557·15-s + 0.343·16-s − 0.888·17-s − 0.162·18-s − 1.50·19-s − 0.0735·20-s − 0.683·21-s − 0.924·22-s + 1.40·23-s + 0.495·24-s − 0.990·25-s − 0.174·26-s + 0.192·27-s + 0.902·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
good | 2 | \( 1 + 2.75T + 32T^{2} \) |
| 5 | \( 1 - 5.39T + 3.12e3T^{2} \) |
| 7 | \( 1 + 153.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 761.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 217.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 538.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.78e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.97e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.80e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 8.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.77e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.47e4T + 8.58e9T^{2} \) |
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
−11.10196019764883245676460308852, −9.960604791953435523353342308357, −8.988817482778570097664652929885, −8.819437871086016907880173796766, −7.13430540505529702871086143761, −6.22372403410979758130240271794, −4.36161309107508631978953191242, −3.49060940722434141366489623569, −1.59982124300523980286507738238, 0,
1.59982124300523980286507738238, 3.49060940722434141366489623569, 4.36161309107508631978953191242, 6.22372403410979758130240271794, 7.13430540505529702871086143761, 8.819437871086016907880173796766, 8.988817482778570097664652929885, 9.960604791953435523353342308357, 11.10196019764883245676460308852