| L(s) = 1 | + 53.0·2-s − 6.75e3·3-s − 2.99e4·4-s − 1.92e5·5-s − 3.58e5·6-s + 1.68e6·7-s − 3.32e6·8-s + 3.12e7·9-s − 1.02e7·10-s + 9.08e7·11-s + 2.02e8·12-s − 1.60e8·13-s + 8.92e7·14-s + 1.30e9·15-s + 8.05e8·16-s − 1.51e9·17-s + 1.65e9·18-s + 1.28e9·19-s + 5.77e9·20-s − 1.13e10·21-s + 4.81e9·22-s + 1.75e7·23-s + 2.24e10·24-s + 6.58e9·25-s − 8.52e9·26-s − 1.14e11·27-s − 5.04e10·28-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 1.78·3-s − 0.914·4-s − 1.10·5-s − 0.522·6-s + 0.772·7-s − 0.560·8-s + 2.17·9-s − 0.322·10-s + 1.40·11-s + 1.63·12-s − 0.711·13-s + 0.226·14-s + 1.96·15-s + 0.750·16-s − 0.893·17-s + 0.638·18-s + 0.330·19-s + 1.00·20-s − 1.37·21-s + 0.411·22-s + 0.00107·23-s + 0.999·24-s + 0.215·25-s − 0.208·26-s − 2.10·27-s − 0.706·28-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 | \( 1 - 53.0T + 3.27e4T^{2} \) |
| 3 | \( 1 + 6.75e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 1.92e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 1.68e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 9.08e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 1.60e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.51e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 1.28e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 1.75e7T + 2.66e20T^{2} \) |
| 31 | \( 1 + 5.75e10T + 2.34e22T^{2} \) |
| 37 | \( 1 - 1.07e12T + 3.33e23T^{2} \) |
| 41 | \( 1 - 6.16e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.84e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 5.22e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.18e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 2.22e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.64e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 2.71e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.43e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.19e14T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.49e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 6.86e12T + 6.11e28T^{2} \) |
| 89 | \( 1 + 1.88e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.41e15T + 6.33e29T^{2} \) |
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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
−12.62022067199815927705607320251, −11.80635475699726960406748093891, −11.04749529943295651158200534423, −9.343916469622114781157484153906, −7.59076940760153520768587244045, −6.11773992493838743057982576032, −4.71000635677834243529904954402, −4.14243571738212928402922311970, −1.04253620663470292790546633274, 0,
1.04253620663470292790546633274, 4.14243571738212928402922311970, 4.71000635677834243529904954402, 6.11773992493838743057982576032, 7.59076940760153520768587244045, 9.343916469622114781157484153906, 11.04749529943295651158200534423, 11.80635475699726960406748093891, 12.62022067199815927705607320251
