| L(s) = 1 | + 4.29·2-s + 27·3-s − 109.·4-s + 241.·5-s + 115.·6-s − 1.02e3·8-s + 729·9-s + 1.03e3·10-s − 6.84e3·11-s − 2.95e3·12-s + 1.03e4·13-s + 6.50e3·15-s + 9.63e3·16-s + 9.43e3·17-s + 3.13e3·18-s + 4.99e4·19-s − 2.64e4·20-s − 2.94e4·22-s − 6.37e4·23-s − 2.75e4·24-s − 2.00e4·25-s + 4.43e4·26-s + 1.96e4·27-s + 1.80e5·29-s + 2.79e4·30-s + 2.79e3·31-s + 1.72e5·32-s + ⋯ |
| L(s) = 1 | + 0.379·2-s + 0.577·3-s − 0.855·4-s + 0.862·5-s + 0.219·6-s − 0.704·8-s + 0.333·9-s + 0.327·10-s − 1.55·11-s − 0.494·12-s + 1.30·13-s + 0.497·15-s + 0.588·16-s + 0.465·17-s + 0.126·18-s + 1.67·19-s − 0.738·20-s − 0.588·22-s − 1.09·23-s − 0.406·24-s − 0.256·25-s + 0.494·26-s + 0.192·27-s + 1.37·29-s + 0.189·30-s + 0.0168·31-s + 0.928·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.913252686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.913252686\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.29T + 128T^{2} \) |
| 5 | \( 1 - 241.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 6.84e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.03e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 9.43e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.99e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.80e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.79e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.07e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.64e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.30e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.20e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.12e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.73e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.07e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.93e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.92e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.76e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.66e6T + 8.07e13T^{2} \) |
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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
−11.98554890810424598072910458015, −10.36564479815906786799547142745, −9.731269969468021168630565014813, −8.611749361949392816676521366668, −7.74887741973930669434339009216, −5.96433996724516851987845910648, −5.19500352343598803466112498112, −3.75020234498035832808852409802, −2.59272904912573961875450659774, −0.937757293998018991731365237905,
0.937757293998018991731365237905, 2.59272904912573961875450659774, 3.75020234498035832808852409802, 5.19500352343598803466112498112, 5.96433996724516851987845910648, 7.74887741973930669434339009216, 8.611749361949392816676521366668, 9.731269969468021168630565014813, 10.36564479815906786799547142745, 11.98554890810424598072910458015
