| L(s) = 1 | − 776.·2-s − 1.91e4·3-s + 7.86e4·4-s + 1.24e6·5-s + 1.48e7·6-s + 1.82e8·7-s + 3.46e8·8-s − 7.95e8·9-s − 9.63e8·10-s + 9.45e9·11-s − 1.50e9·12-s − 4.62e9·13-s − 1.41e11·14-s − 2.37e10·15-s − 3.09e11·16-s − 8.48e11·17-s + 6.18e11·18-s + 1.73e12·19-s + 9.76e10·20-s − 3.48e12·21-s − 7.34e12·22-s − 1.05e13·23-s − 6.62e12·24-s − 1.75e13·25-s + 3.59e12·26-s + 3.74e13·27-s + 1.43e13·28-s + ⋯ |
| L(s) = 1 | − 1.07·2-s − 0.561·3-s + 0.150·4-s + 0.284·5-s + 0.602·6-s + 1.70·7-s + 0.911·8-s − 0.684·9-s − 0.304·10-s + 1.20·11-s − 0.0842·12-s − 0.120·13-s − 1.83·14-s − 0.159·15-s − 1.12·16-s − 1.73·17-s + 0.734·18-s + 1.23·19-s + 0.0426·20-s − 0.958·21-s − 1.29·22-s − 1.22·23-s − 0.511·24-s − 0.919·25-s + 0.129·26-s + 0.945·27-s + 0.256·28-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 | \( 1 + 776.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 1.91e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 1.24e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.82e8T + 1.13e16T^{2} \) |
| 11 | \( 1 - 9.45e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 4.62e9T + 1.46e21T^{2} \) |
| 17 | \( 1 + 8.48e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.73e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.05e13T + 7.46e25T^{2} \) |
| 29 | \( 1 - 1.09e14T + 6.10e27T^{2} \) |
| 31 | \( 1 + 5.48e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 2.54e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 1.16e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 6.25e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 3.57e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 1.21e17T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.12e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 8.78e16T + 4.95e34T^{2} \) |
| 71 | \( 1 - 4.06e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 2.71e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 9.26e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 2.19e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 2.49e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.04e19T + 5.60e37T^{2} \) |
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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
−11.32431949196926858219151156679, −10.03111631574756070231689249943, −8.810453861063490792060889463686, −8.108694958891545043683809778334, −6.71209083216195515815052055607, −5.24890509842958111473437946229, −4.25778979521058992986386571655, −2.03230107991228122176620815940, −1.19739950301810654548212132473, 0,
1.19739950301810654548212132473, 2.03230107991228122176620815940, 4.25778979521058992986386571655, 5.24890509842958111473437946229, 6.71209083216195515815052055607, 8.108694958891545043683809778334, 8.810453861063490792060889463686, 10.03111631574756070231689249943, 11.32431949196926858219151156679
