| L(s) = 1 | + 6.80·2-s − 27·3-s − 81.7·4-s − 12.5·5-s − 183.·6-s − 1.42e3·8-s + 729·9-s − 85.4·10-s + 6.71e3·11-s + 2.20e3·12-s + 8.77e3·13-s + 339.·15-s + 757.·16-s + 1.39e4·17-s + 4.95e3·18-s − 3.43e4·19-s + 1.02e3·20-s + 4.57e4·22-s − 8.45e4·23-s + 3.85e4·24-s − 7.79e4·25-s + 5.96e4·26-s − 1.96e4·27-s − 1.09e5·29-s + 2.30e3·30-s + 1.54e5·31-s + 1.87e5·32-s + ⋯ |
| L(s) = 1 | + 0.601·2-s − 0.577·3-s − 0.638·4-s − 0.0449·5-s − 0.347·6-s − 0.985·8-s + 0.333·9-s − 0.0270·10-s + 1.52·11-s + 0.368·12-s + 1.10·13-s + 0.0259·15-s + 0.0462·16-s + 0.687·17-s + 0.200·18-s − 1.15·19-s + 0.0286·20-s + 0.915·22-s − 1.44·23-s + 0.568·24-s − 0.997·25-s + 0.665·26-s − 0.192·27-s − 0.834·29-s + 0.0155·30-s + 0.929·31-s + 1.01·32-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 6.80T + 128T^{2} \) |
| 5 | \( 1 + 12.5T + 7.81e4T^{2} \) |
| 11 | \( 1 - 6.71e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.77e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.39e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.43e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.45e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.09e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.54e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.53e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.61e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.33e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.56e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.07e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.11e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.24e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.34e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.20e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.18e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.90e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.43e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.18e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.93e6T + 8.07e13T^{2} \) |
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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
−11.57448280044134151035111557849, −10.21606272940204531748726937586, −9.180949587715109989389650779109, −8.150387360604694349537400240631, −6.42427131722655802700267238134, −5.80774480393711159763337863206, −4.32197255365183077130950888078, −3.65223349184123726770275784531, −1.47830218810590571245649195413, 0,
1.47830218810590571245649195413, 3.65223349184123726770275784531, 4.32197255365183077130950888078, 5.80774480393711159763337863206, 6.42427131722655802700267238134, 8.150387360604694349537400240631, 9.180949587715109989389650779109, 10.21606272940204531748726937586, 11.57448280044134151035111557849
