| L(s) = 1 | + 152.·2-s + 729·3-s + 1.51e4·4-s − 3.72e3·5-s + 1.11e5·6-s + 5.89e5·7-s + 1.05e6·8-s + 5.31e5·9-s − 5.69e5·10-s + 5.43e6·11-s + 1.10e7·12-s + 2.28e7·13-s + 9.00e7·14-s − 2.71e6·15-s + 3.74e7·16-s − 1.09e7·17-s + 8.11e7·18-s + 2.75e8·19-s − 5.63e7·20-s + 4.29e8·21-s + 8.29e8·22-s − 7.85e7·23-s + 7.70e8·24-s − 1.20e9·25-s + 3.49e9·26-s + 3.87e8·27-s + 8.91e9·28-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 0.577·3-s + 1.84·4-s − 0.106·5-s + 0.973·6-s + 1.89·7-s + 1.42·8-s + 0.333·9-s − 0.179·10-s + 0.924·11-s + 1.06·12-s + 1.31·13-s + 3.19·14-s − 0.0616·15-s + 0.558·16-s − 0.110·17-s + 0.562·18-s + 1.34·19-s − 0.196·20-s + 1.09·21-s + 1.55·22-s − 0.110·23-s + 0.822·24-s − 0.988·25-s + 2.21·26-s + 0.192·27-s + 3.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(12.50534452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.50534452\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 - 152.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 3.72e3T + 1.22e9T^{2} \) |
| 7 | \( 1 - 5.89e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 5.43e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.28e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.09e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.75e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 7.85e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.24e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + 7.36e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.97e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.27e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.31e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.06e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.42e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 2.06e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 3.95e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 9.27e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.26e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.43e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.65e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 4.12e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 8.48e12T + 6.73e25T^{2} \) |
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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
−10.99309024133946449795145074739, −9.202003311031081280068625953529, −8.114625423731051013673483361293, −7.19556070835668151430216127721, −5.90877241834864070188411710015, −5.02534215440454342776804015297, −4.05437164347678688642649353710, −3.41025699563988051452552566809, −1.93807230427128209455787064614, −1.34242202186277239982575705617,
1.34242202186277239982575705617, 1.93807230427128209455787064614, 3.41025699563988051452552566809, 4.05437164347678688642649353710, 5.02534215440454342776804015297, 5.90877241834864070188411710015, 7.19556070835668151430216127721, 8.114625423731051013673483361293, 9.202003311031081280068625953529, 10.99309024133946449795145074739
