| L(s) = 1 | − 814.·2-s + 2.99e4·3-s + 1.39e5·4-s − 5.47e6·5-s − 2.44e7·6-s + 1.27e8·7-s + 3.13e8·8-s − 2.63e8·9-s + 4.45e9·10-s − 3.99e9·11-s + 4.18e9·12-s − 1.19e10·13-s − 1.04e11·14-s − 1.64e11·15-s − 3.28e11·16-s + 5.20e11·17-s + 2.14e11·18-s − 3.91e11·19-s − 7.64e11·20-s + 3.83e12·21-s + 3.25e12·22-s − 9.09e11·23-s + 9.39e12·24-s + 1.08e13·25-s + 9.70e12·26-s − 4.27e13·27-s + 1.78e13·28-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.879·3-s + 0.266·4-s − 1.25·5-s − 0.989·6-s + 1.19·7-s + 0.825·8-s − 0.226·9-s + 1.41·10-s − 0.510·11-s + 0.234·12-s − 0.311·13-s − 1.34·14-s − 1.10·15-s − 1.19·16-s + 1.06·17-s + 0.255·18-s − 0.278·19-s − 0.334·20-s + 1.05·21-s + 0.574·22-s − 0.105·23-s + 0.725·24-s + 0.570·25-s + 0.350·26-s − 1.07·27-s + 0.319·28-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 | \( 1 + 814.T + 5.24e5T^{2} \) |
| 3 | \( 1 - 2.99e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 5.47e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.27e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 3.99e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 1.19e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 5.20e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 3.91e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 9.09e11T + 7.46e25T^{2} \) |
| 29 | \( 1 - 4.82e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.62e14T + 2.16e28T^{2} \) |
| 37 | \( 1 + 3.23e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.04e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 4.50e14T + 1.08e31T^{2} \) |
| 53 | \( 1 - 2.26e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 5.68e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 7.12e14T + 8.34e33T^{2} \) |
| 67 | \( 1 - 1.66e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 1.66e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 1.94e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 6.18e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 1.27e18T + 2.90e36T^{2} \) |
| 89 | \( 1 + 1.36e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 6.94e18T + 5.60e37T^{2} \) |
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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
−11.07408853553613725218990352237, −9.909500583894846662015145933241, −8.447747662162581964041670085597, −8.175733376993640221703115869190, −7.36357284291638640656430357140, −5.03112772574924032460754284107, −3.86516584316921799071164699211, −2.45882773583831036795292494161, −1.13442543966934538213646740685, 0,
1.13442543966934538213646740685, 2.45882773583831036795292494161, 3.86516584316921799071164699211, 5.03112772574924032460754284107, 7.36357284291638640656430357140, 8.175733376993640221703115869190, 8.447747662162581964041670085597, 9.909500583894846662015145933241, 11.07408853553613725218990352237
