| L(s) = 1 | + 15.7·2-s + 729·3-s − 7.94e3·4-s − 1.82e4·5-s + 1.14e4·6-s − 1.39e5·7-s − 2.54e5·8-s + 5.31e5·9-s − 2.87e5·10-s + 3.23e6·11-s − 5.79e6·12-s − 2.38e7·13-s − 2.19e6·14-s − 1.32e7·15-s + 6.10e7·16-s − 1.01e8·17-s + 8.37e6·18-s − 2.58e8·19-s + 1.44e8·20-s − 1.01e8·21-s + 5.09e7·22-s + 4.75e8·23-s − 1.85e8·24-s − 8.88e8·25-s − 3.76e8·26-s + 3.87e8·27-s + 1.10e9·28-s + ⋯ |
| L(s) = 1 | + 0.174·2-s + 0.577·3-s − 0.969·4-s − 0.521·5-s + 0.100·6-s − 0.448·7-s − 0.343·8-s + 0.333·9-s − 0.0908·10-s + 0.549·11-s − 0.559·12-s − 1.37·13-s − 0.0780·14-s − 0.301·15-s + 0.909·16-s − 1.01·17-s + 0.0580·18-s − 1.26·19-s + 0.506·20-s − 0.258·21-s + 0.0957·22-s + 0.669·23-s − 0.198·24-s − 0.727·25-s − 0.238·26-s + 0.192·27-s + 0.434·28-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
| \(L(7)\) |
\(\approx\) |
\(0.5936686115\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5936686115\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 - 15.7T + 8.19e3T^{2} \) |
| 5 | \( 1 + 1.82e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 1.39e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 3.23e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 2.38e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.01e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.58e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 4.75e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 4.16e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 3.82e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.63e9T + 2.43e20T^{2} \) |
| 41 | \( 1 + 1.28e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 9.65e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 8.50e9T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.27e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 2.60e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 5.68e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 7.25e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 8.77e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 1.57e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.61e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 8.27e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.71e12T + 6.73e25T^{2} \) |
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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
−10.04044379629846404519557031169, −9.283059132254533836110477468768, −8.489274000649385139109860472454, −7.45385857063942125877735875156, −6.33956708675538026891387128199, −4.84741253377822086659692673406, −4.14095092925999023603112372785, −3.15514706900656458552014565997, −1.91444697253846284949381996826, −0.29721261984394390565800647306,
0.29721261984394390565800647306, 1.91444697253846284949381996826, 3.15514706900656458552014565997, 4.14095092925999023603112372785, 4.84741253377822086659692673406, 6.33956708675538026891387128199, 7.45385857063942125877735875156, 8.489274000649385139109860472454, 9.283059132254533836110477468768, 10.04044379629846404519557031169
