| L(s) = 1 | − 12.8·2-s − 204.·3-s − 346.·4-s + 120.·5-s + 2.63e3·6-s + 4.52e3·7-s + 1.10e4·8-s + 2.22e4·9-s − 1.55e3·10-s − 5.78e4·11-s + 7.09e4·12-s + 5.07e4·13-s − 5.81e4·14-s − 2.47e4·15-s + 3.55e4·16-s − 2.85e5·18-s − 8.15e5·19-s − 4.18e4·20-s − 9.26e5·21-s + 7.44e5·22-s + 7.45e5·23-s − 2.26e6·24-s − 1.93e6·25-s − 6.53e5·26-s − 5.21e5·27-s − 1.56e6·28-s + 1.01e6·29-s + ⋯ |
| L(s) = 1 | − 0.568·2-s − 1.45·3-s − 0.677·4-s + 0.0863·5-s + 0.829·6-s + 0.712·7-s + 0.953·8-s + 1.12·9-s − 0.0490·10-s − 1.19·11-s + 0.988·12-s + 0.493·13-s − 0.404·14-s − 0.125·15-s + 0.135·16-s − 0.641·18-s − 1.43·19-s − 0.0584·20-s − 1.03·21-s + 0.677·22-s + 0.555·23-s − 1.39·24-s − 0.992·25-s − 0.280·26-s − 0.188·27-s − 0.482·28-s + 0.265·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 12.8T + 512T^{2} \) |
| 3 | \( 1 + 204.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 120.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 4.52e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.78e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 5.07e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 8.15e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.45e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.01e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.30e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.57e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.94e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.64e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.34e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.09e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.72e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.80e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.00e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.02e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.38e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.10e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.30e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.52e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.25e8T + 7.60e17T^{2} \) |
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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
−10.05541767602545997975320839777, −8.721241971047435184725997959385, −7.985573167198169807826487536480, −6.82092185564247093473272042176, −5.58469830499905913878761749558, −5.02060150280409146777323888651, −4.01471551746692890294161270870, −2.02324166833455156801454908166, −0.815918434021225004181743134585, 0,
0.815918434021225004181743134585, 2.02324166833455156801454908166, 4.01471551746692890294161270870, 5.02060150280409146777323888651, 5.58469830499905913878761749558, 6.82092185564247093473272042176, 7.985573167198169807826487536480, 8.721241971047435184725997959385, 10.05541767602545997975320839777
