| L(s) = 1 | − 12.1·2-s + 27·3-s + 19.3·4-s + 236.·5-s − 327.·6-s + 1.42e3·7-s + 1.31e3·8-s + 729·9-s − 2.86e3·10-s − 5.47e3·11-s + 522.·12-s − 8.45e3·13-s − 1.73e4·14-s + 6.38e3·15-s − 1.84e4·16-s − 6.08e3·17-s − 8.84e3·18-s − 1.29e4·19-s + 4.57e3·20-s + 3.85e4·21-s + 6.64e4·22-s − 5.39e4·23-s + 3.56e4·24-s − 2.22e4·25-s + 1.02e5·26-s + 1.96e4·27-s + 2.76e4·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s + 0.577·3-s + 0.151·4-s + 0.845·5-s − 0.619·6-s + 1.57·7-s + 0.910·8-s + 0.333·9-s − 0.907·10-s − 1.23·11-s + 0.0873·12-s − 1.06·13-s − 1.68·14-s + 0.488·15-s − 1.12·16-s − 0.300·17-s − 0.357·18-s − 0.433·19-s + 0.127·20-s + 0.907·21-s + 1.33·22-s − 0.925·23-s + 0.525·24-s − 0.285·25-s + 1.14·26-s + 0.192·27-s + 0.237·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 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
−10.39966294614909563839860067099, −9.945874034494061471558135418044, −8.735681492055786641154260981859, −8.059590367850017130745347951633, −7.27592121809788550627796916453, −5.39064571960200604360136255650, −4.48496486458400643315626290780, −2.30854229768775414944186637012, −1.65972904065695586245002404394, 0,
1.65972904065695586245002404394, 2.30854229768775414944186637012, 4.48496486458400643315626290780, 5.39064571960200604360136255650, 7.27592121809788550627796916453, 8.059590367850017130745347951633, 8.735681492055786641154260981859, 9.945874034494061471558135418044, 10.39966294614909563839860067099
