| L(s) = 1 | + 6.01·2-s − 91.8·4-s + 385.·5-s + 847.·7-s − 1.32e3·8-s + 2.32e3·10-s + 5.52e3·11-s + 2.98e3·13-s + 5.09e3·14-s + 3.79e3·16-s + 6.65e3·17-s + 4.74e4·19-s − 3.54e4·20-s + 3.32e4·22-s − 1.69e4·23-s + 7.07e4·25-s + 1.79e4·26-s − 7.78e4·28-s − 2.63e3·29-s + 1.38e5·31-s + 1.92e5·32-s + 4.00e4·34-s + 3.27e5·35-s − 9.12e4·37-s + 2.85e5·38-s − 5.10e5·40-s − 5.49e5·41-s + ⋯ |
| L(s) = 1 | + 0.531·2-s − 0.717·4-s + 1.38·5-s + 0.934·7-s − 0.913·8-s + 0.733·10-s + 1.25·11-s + 0.377·13-s + 0.496·14-s + 0.231·16-s + 0.328·17-s + 1.58·19-s − 0.990·20-s + 0.665·22-s − 0.289·23-s + 0.905·25-s + 0.200·26-s − 0.670·28-s − 0.0200·29-s + 0.836·31-s + 1.03·32-s + 0.174·34-s + 1.28·35-s − 0.296·37-s + 0.843·38-s − 1.26·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.613869580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.613869580\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 - 6.01T + 128T^{2} \) |
| 5 | \( 1 - 385.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 847.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.52e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.98e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.65e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.74e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.69e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.63e3T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.38e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 9.12e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.49e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.40e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.17e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.00e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 7.14e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 9.12e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.74e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.34e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.12e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.95e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.69e6T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.581091357983498822896560783600, −9.016685845300106981243433739000, −8.069343349235523069442252716933, −6.72681297719858340332880470623, −5.76419560369578070183892446017, −5.18741532059420825150603962231, −4.17359949770006719696261434222, −3.09331740522763421465591664076, −1.69298751127431239938567328625, −0.961940067735263229870880578500,
0.961940067735263229870880578500, 1.69298751127431239938567328625, 3.09331740522763421465591664076, 4.17359949770006719696261434222, 5.18741532059420825150603962231, 5.76419560369578070183892446017, 6.72681297719858340332880470623, 8.069343349235523069442252716933, 9.016685845300106981243433739000, 9.581091357983498822896560783600
