| L(s) = 1 | − 235.·2-s + 964.·3-s − 7.56e4·4-s + 8.64e5·5-s − 2.27e5·6-s + 6.42e6·7-s + 4.86e7·8-s − 1.28e8·9-s − 2.03e8·10-s − 1.22e9·11-s − 7.30e7·12-s + 3.51e9·13-s − 1.51e9·14-s + 8.34e8·15-s − 1.52e9·16-s + 2.92e10·17-s + 3.01e10·18-s − 1.20e11·19-s − 6.54e10·20-s + 6.20e9·21-s + 2.88e11·22-s + 6.15e11·23-s + 4.69e10·24-s − 1.55e10·25-s − 8.27e11·26-s − 2.48e11·27-s − 4.86e11·28-s + ⋯ |
| L(s) = 1 | − 0.649·2-s + 0.0849·3-s − 0.577·4-s + 0.989·5-s − 0.0551·6-s + 0.421·7-s + 1.02·8-s − 0.992·9-s − 0.643·10-s − 1.72·11-s − 0.0490·12-s + 1.19·13-s − 0.273·14-s + 0.0840·15-s − 0.0889·16-s + 1.01·17-s + 0.645·18-s − 1.62·19-s − 0.571·20-s + 0.0357·21-s + 1.12·22-s + 1.63·23-s + 0.0870·24-s − 0.0204·25-s − 0.776·26-s − 0.169·27-s − 0.243·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 5.00e11T \) |
| good | 2 | \( 1 + 235.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 964.T + 1.29e8T^{2} \) |
| 5 | \( 1 - 8.64e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 6.42e6T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.22e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.51e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 2.92e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.20e11T + 5.48e21T^{2} \) |
| 23 | \( 1 - 6.15e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 7.49e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.39e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 1.24e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 5.66e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.16e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 1.72e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.37e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 4.28e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 5.49e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.52e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.29e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.35e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.76e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 2.31e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 6.79e16T + 5.95e33T^{2} \) |
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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
−13.05844572384388369836889696254, −10.97633407527032049553759246359, −10.11927470972151705415316605657, −8.742364937123710623319969134012, −8.006558033483086972487768775572, −5.95646150886012356535876952675, −4.87043673638856176630403781355, −2.86222493093115302305482694498, −1.40318828556377803718861581209, 0,
1.40318828556377803718861581209, 2.86222493093115302305482694498, 4.87043673638856176630403781355, 5.95646150886012356535876952675, 8.006558033483086972487768775572, 8.742364937123710623319969134012, 10.11927470972151705415316605657, 10.97633407527032049553759246359, 13.05844572384388369836889696254
