| L(s) = 1 | − 1.32e3·2-s + 1.44e4·3-s + 1.23e6·4-s − 3.85e6·5-s − 1.91e7·6-s − 1.85e8·7-s − 9.48e8·8-s − 9.54e8·9-s + 5.12e9·10-s − 1.54e10·11-s + 1.78e10·12-s + 6.68e9·13-s + 2.45e11·14-s − 5.56e10·15-s + 6.09e11·16-s + 3.69e11·17-s + 1.26e12·18-s + 1.44e12·19-s − 4.77e12·20-s − 2.67e12·21-s + 2.04e13·22-s + 1.37e13·23-s − 1.36e13·24-s − 4.18e12·25-s − 8.87e12·26-s − 3.05e13·27-s − 2.29e14·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.422·3-s + 2.36·4-s − 0.883·5-s − 0.775·6-s − 1.73·7-s − 2.49·8-s − 0.821·9-s + 1.61·10-s − 1.97·11-s + 0.998·12-s + 0.174·13-s + 3.18·14-s − 0.373·15-s + 2.21·16-s + 0.754·17-s + 1.50·18-s + 1.02·19-s − 2.08·20-s − 0.733·21-s + 3.61·22-s + 1.58·23-s − 1.05·24-s − 0.219·25-s − 0.320·26-s − 0.770·27-s − 4.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 - 1.11e15T \) |
| good | 2 | \( 1 + 1.32e3T + 5.24e5T^{2} \) |
| 3 | \( 1 - 1.44e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 3.85e6T + 1.90e13T^{2} \) |
| 7 | \( 1 + 1.85e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 1.54e10T + 6.11e19T^{2} \) |
| 13 | \( 1 - 6.68e9T + 1.46e21T^{2} \) |
| 17 | \( 1 - 3.69e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.44e12T + 1.97e24T^{2} \) |
| 23 | \( 1 - 1.37e13T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.08e14T + 6.10e27T^{2} \) |
| 31 | \( 1 + 7.49e13T + 2.16e28T^{2} \) |
| 37 | \( 1 - 4.46e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 5.84e14T + 4.39e30T^{2} \) |
| 43 | \( 1 - 1.13e15T + 1.08e31T^{2} \) |
| 53 | \( 1 - 8.46e15T + 5.77e32T^{2} \) |
| 59 | \( 1 - 8.21e16T + 4.42e33T^{2} \) |
| 61 | \( 1 + 8.37e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.80e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 3.85e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 4.68e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 7.10e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.59e17T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.14e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.07e17T + 5.60e37T^{2} \) |
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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
−10.80273417830440501374113778239, −9.909561202985733379808625752590, −8.927096536421243839286229000808, −7.909932092334622069533135976702, −7.18895572230348823522542548280, −5.74125114125024708271785081942, −3.06942086926033460816703290328, −2.80174207256548464917368678840, −0.74904656989753533004248175507, 0,
0.74904656989753533004248175507, 2.80174207256548464917368678840, 3.06942086926033460816703290328, 5.74125114125024708271785081942, 7.18895572230348823522542548280, 7.909932092334622069533135976702, 8.927096536421243839286229000808, 9.909561202985733379808625752590, 10.80273417830440501374113778239
