| L(s) = 1 | − 273.·2-s + 4.14e3·3-s + 4.21e4·4-s − 1.42e5·5-s − 1.13e6·6-s − 8.23e5·7-s − 2.56e6·8-s + 2.86e6·9-s + 3.90e7·10-s + 8.52e7·11-s + 1.74e8·12-s + 4.02e7·13-s + 2.25e8·14-s − 5.92e8·15-s − 6.78e8·16-s + 2.32e9·17-s − 7.83e8·18-s + 5.71e9·19-s − 6.01e9·20-s − 3.41e9·21-s − 2.33e10·22-s + 2.36e10·23-s − 1.06e10·24-s − 1.01e10·25-s − 1.10e10·26-s − 4.76e10·27-s − 3.47e10·28-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.09·3-s + 1.28·4-s − 0.817·5-s − 1.65·6-s − 0.377·7-s − 0.432·8-s + 0.199·9-s + 1.23·10-s + 1.31·11-s + 1.40·12-s + 0.178·13-s + 0.571·14-s − 0.895·15-s − 0.632·16-s + 1.37·17-s − 0.301·18-s + 1.46·19-s − 1.05·20-s − 0.413·21-s − 1.99·22-s + 1.44·23-s − 0.473·24-s − 0.331·25-s − 0.269·26-s − 0.876·27-s − 0.486·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.057820143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.057820143\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 8.23e5T \) |
| good | 2 | \( 1 + 273.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 4.14e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 1.42e5T + 3.05e10T^{2} \) |
| 11 | \( 1 - 8.52e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 4.02e7T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.32e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 5.71e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.36e10T + 2.66e20T^{2} \) |
| 29 | \( 1 - 8.76e10T + 8.62e21T^{2} \) |
| 31 | \( 1 - 6.15e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 5.62e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.33e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.20e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 4.05e11T + 1.20e25T^{2} \) |
| 53 | \( 1 + 8.95e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 3.28e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.49e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 4.30e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 7.86e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.05e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.66e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 1.96e13T + 6.11e28T^{2} \) |
| 89 | \( 1 - 3.89e12T + 1.74e29T^{2} \) |
| 97 | \( 1 - 6.49e14T + 6.33e29T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.97096020378321124426600873027, −17.14292534824713527884914025554, −15.77246013799156799535886135592, −14.14953490530136642896293403514, −11.69711082189051883391221989743, −9.688199349244555659532032088179, −8.570230134893529304300717938060, −7.30597347295837056928763103445, −3.32239275117640040737962710298, −1.06182631420308425628775748045,
1.06182631420308425628775748045, 3.32239275117640040737962710298, 7.30597347295837056928763103445, 8.570230134893529304300717938060, 9.688199349244555659532032088179, 11.69711082189051883391221989743, 14.14953490530136642896293403514, 15.77246013799156799535886135592, 17.14292534824713527884914025554, 18.97096020378321124426600873027
