| L(s) = 1 | − 120.·2-s − 729·3-s + 6.21e3·4-s + 2.65e4·5-s + 8.74e4·6-s − 2.50e5·7-s + 2.37e5·8-s + 5.31e5·9-s − 3.19e6·10-s − 6.39e6·11-s − 4.52e6·12-s + 6.37e6·13-s + 3.01e7·14-s − 1.93e7·15-s − 7.94e7·16-s − 1.01e8·17-s − 6.37e7·18-s − 4.01e7·19-s + 1.65e8·20-s + 1.82e8·21-s + 7.67e8·22-s − 1.26e9·23-s − 1.73e8·24-s − 5.14e8·25-s − 7.65e8·26-s − 3.87e8·27-s − 1.55e9·28-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 0.577·3-s + 0.758·4-s + 0.760·5-s + 0.765·6-s − 0.805·7-s + 0.320·8-s + 0.333·9-s − 1.00·10-s − 1.08·11-s − 0.437·12-s + 0.366·13-s + 1.06·14-s − 0.439·15-s − 1.18·16-s − 1.01·17-s − 0.442·18-s − 0.195·19-s + 0.576·20-s + 0.465·21-s + 1.44·22-s − 1.77·23-s − 0.185·24-s − 0.421·25-s − 0.485·26-s − 0.192·27-s − 0.611·28-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
| \(L(7)\) |
\(\approx\) |
\(0.07572903526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07572903526\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 120.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 2.65e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.50e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 6.39e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 6.37e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.01e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 4.01e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.26e9T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.74e7T + 1.02e19T^{2} \) |
| 31 | \( 1 + 2.60e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.07e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.06e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.18e8T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.23e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.89e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 4.47e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 7.76e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 4.40e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.80e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 4.23e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 3.86e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 1.27e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.01e13T + 6.73e25T^{2} \) |
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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
−10.26111945783764426238534909585, −9.478210010193324498074895769118, −8.534243711340274545951771902921, −7.45412387957406816548797568021, −6.41689452684989918782302107156, −5.53490962059890396871587689364, −4.13863896396678916331631880439, −2.44388721302063844620364604582, −1.57831360778043825399738866260, −0.14563582535774500132665508459,
0.14563582535774500132665508459, 1.57831360778043825399738866260, 2.44388721302063844620364604582, 4.13863896396678916331631880439, 5.53490962059890396871587689364, 6.41689452684989918782302107156, 7.45412387957406816548797568021, 8.534243711340274545951771902921, 9.478210010193324498074895769118, 10.26111945783764426238534909585
