| L(s) = 1 | − 310.·2-s − 3.40e3·3-s + 6.37e4·4-s − 2.28e5·5-s + 1.05e6·6-s − 4.99e5·7-s − 9.63e6·8-s − 2.72e6·9-s + 7.10e7·10-s − 2.17e7·11-s − 2.17e8·12-s − 4.18e8·13-s + 1.55e8·14-s + 7.79e8·15-s + 9.04e8·16-s + 1.37e9·17-s + 8.46e8·18-s + 4.88e9·19-s − 1.45e10·20-s + 1.70e9·21-s + 6.75e9·22-s + 2.70e10·23-s + 3.28e10·24-s + 2.17e10·25-s + 1.30e11·26-s + 5.82e10·27-s − 3.18e10·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.900·3-s + 1.94·4-s − 1.30·5-s + 1.54·6-s − 0.229·7-s − 1.62·8-s − 0.189·9-s + 2.24·10-s − 0.336·11-s − 1.75·12-s − 1.85·13-s + 0.393·14-s + 1.17·15-s + 0.842·16-s + 0.814·17-s + 0.325·18-s + 1.25·19-s − 2.54·20-s + 0.206·21-s + 0.576·22-s + 1.65·23-s + 1.46·24-s + 0.713·25-s + 3.17·26-s + 1.07·27-s − 0.446·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 1.72e10T \) |
| good | 2 | \( 1 + 310.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 3.40e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 2.28e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 4.99e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 2.17e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 4.18e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 1.37e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 4.88e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.70e10T + 2.66e20T^{2} \) |
| 31 | \( 1 + 1.32e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 1.33e10T + 3.33e23T^{2} \) |
| 41 | \( 1 + 5.86e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.11e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 4.68e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 9.23e11T + 7.31e25T^{2} \) |
| 59 | \( 1 - 1.94e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.75e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 1.55e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.02e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 6.43e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.93e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 3.27e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 2.02e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 4.89e13T + 6.33e29T^{2} \) |
| 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
−12.25185931612636657795071929680, −11.55155287614296730220626436545, −10.49290335894015142796613539116, −9.254395275115734696876341279962, −7.76157088120929838025773335103, −7.09777659974754237688316432579, −5.15726957226645236356873793904, −2.89653730229620046251490715508, −0.824777580503130726869400207370, 0,
0.824777580503130726869400207370, 2.89653730229620046251490715508, 5.15726957226645236356873793904, 7.09777659974754237688316432579, 7.76157088120929838025773335103, 9.254395275115734696876341279962, 10.49290335894015142796613539116, 11.55155287614296730220626436545, 12.25185931612636657795071929680
