| L(s) = 1 | − 4.75·2-s − 9.40·4-s + 8.06·5-s − 28.8·7-s + 196.·8-s − 38.3·10-s − 556.·11-s − 1.12e3·13-s + 137.·14-s − 634.·16-s − 447.·17-s − 1.35e3·19-s − 75.8·20-s + 2.64e3·22-s + 4.46e3·23-s − 3.05e3·25-s + 5.36e3·26-s + 271.·28-s + 2.94e3·29-s + 1.11e3·31-s − 3.28e3·32-s + 2.12e3·34-s − 232.·35-s − 5.87e3·37-s + 6.41e3·38-s + 1.58e3·40-s + 638.·41-s + ⋯ |
| L(s) = 1 | − 0.840·2-s − 0.293·4-s + 0.144·5-s − 0.222·7-s + 1.08·8-s − 0.121·10-s − 1.38·11-s − 1.85·13-s + 0.187·14-s − 0.619·16-s − 0.375·17-s − 0.858·19-s − 0.0424·20-s + 1.16·22-s + 1.76·23-s − 0.979·25-s + 1.55·26-s + 0.0653·28-s + 0.649·29-s + 0.208·31-s − 0.566·32-s + 0.315·34-s − 0.0321·35-s − 0.705·37-s + 0.721·38-s + 0.156·40-s + 0.0593·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2915638855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2915638855\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 + 4.75T + 32T^{2} \) |
| 5 | \( 1 - 8.06T + 3.12e3T^{2} \) |
| 7 | \( 1 + 28.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 556.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 447.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.46e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 638.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.93e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 7.81e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.20e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.62e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.83e4T + 8.58e9T^{2} \) |
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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
−10.02127723577823625588218878041, −9.257139283071299506388737743057, −8.297025697967626025231952937070, −7.58039278090423737938364893972, −6.69763931582661264147996983108, −5.13850092928610614420249968785, −4.67525711175831331956039474856, −2.98080657159775588057388686003, −1.90438605640333905132727412974, −0.29313114260575161532681701445,
0.29313114260575161532681701445, 1.90438605640333905132727412974, 2.98080657159775588057388686003, 4.67525711175831331956039474856, 5.13850092928610614420249968785, 6.69763931582661264147996983108, 7.58039278090423737938364893972, 8.297025697967626025231952937070, 9.257139283071299506388737743057, 10.02127723577823625588218878041
