| L(s) = 1 | + 12.1·2-s + 19.3·4-s − 236.·5-s + 1.42e3·7-s − 1.31e3·8-s − 2.86e3·10-s + 5.47e3·11-s − 8.45e3·13-s + 1.73e4·14-s − 1.84e4·16-s + 6.08e3·17-s − 1.29e4·19-s − 4.57e3·20-s + 6.64e4·22-s + 5.39e4·23-s − 2.22e4·25-s − 1.02e5·26-s + 2.76e4·28-s + 1.43e4·29-s − 4.84e4·31-s − 5.56e4·32-s + 7.38e4·34-s − 3.37e5·35-s + 8.28e4·37-s − 1.57e5·38-s + 3.11e5·40-s + 7.82e5·41-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.151·4-s − 0.845·5-s + 1.57·7-s − 0.910·8-s − 0.907·10-s + 1.23·11-s − 1.06·13-s + 1.68·14-s − 1.12·16-s + 0.300·17-s − 0.433·19-s − 0.127·20-s + 1.33·22-s + 0.925·23-s − 0.285·25-s − 1.14·26-s + 0.237·28-s + 0.109·29-s − 0.292·31-s − 0.299·32-s + 0.322·34-s − 1.32·35-s + 0.269·37-s − 0.465·38-s + 0.770·40-s + 1.77·41-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
| \(L(4)\) |
\(\approx\) |
\(3.509316315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.509316315\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 12.1T + 128T^{2} \) |
| 5 | \( 1 + 236.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.42e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.47e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.45e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.08e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.29e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.39e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.43e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.84e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 8.28e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 7.82e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.69e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.68e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.36e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 3.73e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.64e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.64e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.63e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 7.17e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.06e7T + 8.07e13T^{2} \) |
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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
−9.604039936226085402350340211910, −8.711844479232959858762158583553, −7.84829980838393717066508860538, −6.95969598903545628460500144245, −5.75499031178273349502217104875, −4.71518435606599876628924331534, −4.34132858279796193295116843136, −3.30664027907559684805221561077, −1.99180000255376913210499705975, −0.69896490452828524142793584359,
0.69896490452828524142793584359, 1.99180000255376913210499705975, 3.30664027907559684805221561077, 4.34132858279796193295116843136, 4.71518435606599876628924331534, 5.75499031178273349502217104875, 6.95969598903545628460500144245, 7.84829980838393717066508860538, 8.711844479232959858762158583553, 9.604039936226085402350340211910
