| L(s) = 1 | − 1.07e3·2-s − 3.31e4·3-s + 6.31e5·4-s − 6.46e6·5-s + 3.56e7·6-s + 1.13e7·7-s − 1.15e8·8-s − 6.54e7·9-s + 6.95e9·10-s + 3.37e9·11-s − 2.09e10·12-s + 1.74e10·13-s − 1.22e10·14-s + 2.14e11·15-s − 2.06e11·16-s + 6.53e11·17-s + 7.04e10·18-s + 1.81e12·19-s − 4.08e12·20-s − 3.76e11·21-s − 3.62e12·22-s − 3.75e12·23-s + 3.83e12·24-s + 2.27e13·25-s − 1.87e13·26-s + 4.06e13·27-s + 7.19e12·28-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 0.971·3-s + 1.20·4-s − 1.48·5-s + 1.44·6-s + 0.106·7-s − 0.304·8-s − 0.0563·9-s + 2.19·10-s + 0.431·11-s − 1.17·12-s + 0.455·13-s − 0.158·14-s + 1.43·15-s − 0.752·16-s + 1.33·17-s + 0.0836·18-s + 1.29·19-s − 1.78·20-s − 0.103·21-s − 0.640·22-s − 0.434·23-s + 0.296·24-s + 1.19·25-s − 0.676·26-s + 1.02·27-s + 0.128·28-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
| \(L(10)\) |
\(\approx\) |
\(0.4856684484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4856684484\) |
| \(L(\frac{21}{2})\) |
|
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 | \( 1 + 1.07e3T + 5.24e5T^{2} \) |
| 3 | \( 1 + 3.31e4T + 1.16e9T^{2} \) |
| 5 | \( 1 + 6.46e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 1.13e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 3.37e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 1.74e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 6.53e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 1.81e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 3.75e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 3.07e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.81e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.15e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 2.53e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 1.68e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 4.77e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 7.45e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 3.07e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 1.03e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 8.21e16T + 1.49e35T^{2} \) |
| 73 | \( 1 - 1.81e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 9.79e17T + 1.13e36T^{2} \) |
| 83 | \( 1 - 1.82e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 6.01e17T + 1.09e37T^{2} \) |
| 97 | \( 1 - 1.83e18T + 5.60e37T^{2} \) |
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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
−11.57204395152504047228201037352, −10.75308759321436066685938164305, −9.524279534679910738215580306332, −8.183224087731853134894201193242, −7.57290439239558836326695721819, −6.24082913592492861269456091155, −4.69025952808631782207621924855, −3.23139049479116573165707616825, −1.16046848260468808368486579682, −0.54417458893701877967175281898,
0.54417458893701877967175281898, 1.16046848260468808368486579682, 3.23139049479116573165707616825, 4.69025952808631782207621924855, 6.24082913592492861269456091155, 7.57290439239558836326695721819, 8.183224087731853134894201193242, 9.524279534679910738215580306332, 10.75308759321436066685938164305, 11.57204395152504047228201037352
