| L(s) = 1 | − 289.·2-s + 4.72e3·3-s + 5.10e4·4-s + 4.27e4·5-s − 1.36e6·6-s + 3.84e6·7-s − 5.28e6·8-s + 7.99e6·9-s − 1.23e7·10-s − 9.93e7·11-s + 2.41e8·12-s − 2.12e8·13-s − 1.11e9·14-s + 2.02e8·15-s − 1.41e8·16-s − 1.04e9·17-s − 2.31e9·18-s − 7.23e9·19-s + 2.18e9·20-s + 1.81e10·21-s + 2.87e10·22-s + 2.50e10·23-s − 2.49e10·24-s − 2.86e10·25-s + 6.14e10·26-s − 3.00e10·27-s + 1.96e11·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 1.24·3-s + 1.55·4-s + 0.244·5-s − 1.99·6-s + 1.76·7-s − 0.891·8-s + 0.556·9-s − 0.391·10-s − 1.53·11-s + 1.94·12-s − 0.937·13-s − 2.82·14-s + 0.305·15-s − 0.132·16-s − 0.615·17-s − 0.890·18-s − 1.85·19-s + 0.381·20-s + 2.20·21-s + 2.45·22-s + 1.53·23-s − 1.11·24-s − 0.940·25-s + 1.49·26-s − 0.552·27-s + 2.74·28-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 | \( 1 + 289.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 4.72e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 4.27e4T + 3.05e10T^{2} \) |
| 7 | \( 1 - 3.84e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 9.93e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 2.12e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.04e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 7.23e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.50e10T + 2.66e20T^{2} \) |
| 31 | \( 1 + 6.48e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 6.94e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.24e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 2.69e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 2.08e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 3.67e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 2.50e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.69e12T + 6.02e26T^{2} \) |
| 67 | \( 1 + 1.14e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 7.22e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 2.46e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 6.45e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 9.53e13T + 6.11e28T^{2} \) |
| 89 | \( 1 + 3.92e13T + 1.74e29T^{2} \) |
| 97 | \( 1 - 7.04e14T + 6.33e29T^{2} \) |
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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
−13.23187578475276568065421179525, −11.20469677862908298172345366590, −10.27012283028818313109031352951, −8.833187321464704149250503721829, −8.209864081147876271168873937388, −7.35545925286665782390344837972, −4.84696045471246976193625527671, −2.42456142204521349513330700938, −1.83979649443572511872230551893, 0,
1.83979649443572511872230551893, 2.42456142204521349513330700938, 4.84696045471246976193625527671, 7.35545925286665782390344837972, 8.209864081147876271168873937388, 8.833187321464704149250503721829, 10.27012283028818313109031352951, 11.20469677862908298172345366590, 13.23187578475276568065421179525
