| L(s) = 1 | − 151.·2-s − 729·3-s + 1.46e4·4-s + 6.49e4·5-s + 1.10e5·6-s − 3.52e4·7-s − 9.82e5·8-s + 5.31e5·9-s − 9.81e6·10-s − 8.46e5·11-s − 1.07e7·12-s − 6.32e6·13-s + 5.33e6·14-s − 4.73e7·15-s + 2.82e7·16-s + 8.25e7·17-s − 8.03e7·18-s + 3.86e8·19-s + 9.53e8·20-s + 2.56e7·21-s + 1.28e8·22-s − 5.52e8·23-s + 7.16e8·24-s + 2.99e9·25-s + 9.56e8·26-s − 3.87e8·27-s − 5.17e8·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s − 0.577·3-s + 1.79·4-s + 1.85·5-s + 0.964·6-s − 0.113·7-s − 1.32·8-s + 0.333·9-s − 3.10·10-s − 0.144·11-s − 1.03·12-s − 0.363·13-s + 0.189·14-s − 1.07·15-s + 0.421·16-s + 0.829·17-s − 0.557·18-s + 1.88·19-s + 3.33·20-s + 0.0653·21-s + 0.240·22-s − 0.778·23-s + 0.765·24-s + 2.45·25-s + 0.607·26-s − 0.192·27-s − 0.202·28-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
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 151.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 6.49e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.52e4T + 9.68e10T^{2} \) |
| 11 | \( 1 + 8.46e5T + 3.45e13T^{2} \) |
| 13 | \( 1 + 6.32e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 8.25e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 3.86e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 5.52e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.29e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 1.19e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.10e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 5.49e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.86e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.24e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.31e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 2.65e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 6.42e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.49e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 8.05e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 2.72e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 5.25e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 2.96e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.05e13T + 6.73e25T^{2} \) |
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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
−9.631989699251795270677691185321, −9.400841105241315842412935431734, −7.917412917305559253464362806548, −6.96733828064004674136764912075, −5.94818711133514438443349360677, −5.19494634284328060470056506449, −2.95446819708700209020971651728, −1.76270888950643226256411007990, −1.22591892734187600314610710585, 0,
1.22591892734187600314610710585, 1.76270888950643226256411007990, 2.95446819708700209020971651728, 5.19494634284328060470056506449, 5.94818711133514438443349360677, 6.96733828064004674136764912075, 7.917412917305559253464362806548, 9.400841105241315842412935431734, 9.631989699251795270677691185321
