| L(s) = 1 | + 354.·2-s + 6.64e3·3-s + 9.30e4·4-s − 1.72e5·5-s + 2.35e6·6-s + 4.17e5·7-s + 2.14e7·8-s + 2.98e7·9-s − 6.10e7·10-s − 2.29e7·11-s + 6.18e8·12-s − 4.10e8·13-s + 1.48e8·14-s − 1.14e9·15-s + 4.54e9·16-s + 2.48e9·17-s + 1.05e10·18-s + 1.32e9·19-s − 1.60e10·20-s + 2.77e9·21-s − 8.13e9·22-s − 9.72e9·23-s + 1.42e11·24-s − 8.80e8·25-s − 1.45e11·26-s + 1.03e11·27-s + 3.88e10·28-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + 1.75·3-s + 2.84·4-s − 0.985·5-s + 3.43·6-s + 0.191·7-s + 3.60·8-s + 2.08·9-s − 1.93·10-s − 0.354·11-s + 4.98·12-s − 1.81·13-s + 0.375·14-s − 1.72·15-s + 4.23·16-s + 1.46·17-s + 4.07·18-s + 0.339·19-s − 2.79·20-s + 0.336·21-s − 0.695·22-s − 0.595·23-s + 6.33·24-s − 0.0288·25-s − 3.56·26-s + 1.89·27-s + 0.544·28-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
| \(L(8)\) |
\(\approx\) |
\(11.27308813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.27308813\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 - 354.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 6.64e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 1.72e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 4.17e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 2.29e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 4.10e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 2.48e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 1.32e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 9.72e9T + 2.66e20T^{2} \) |
| 31 | \( 1 + 1.20e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 7.54e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.85e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.61e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 9.07e11T + 1.20e25T^{2} \) |
| 53 | \( 1 - 4.99e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 2.82e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 2.96e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 3.59e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.50e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 9.09e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.62e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.84e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 2.19e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 8.06e14T + 6.33e29T^{2} \) |
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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
−13.98937788657479700730583424128, −12.68917543023594488072804297644, −11.90628051531345037598872307487, −10.08078257358470524399030262779, −7.80978734107613129448963290835, −7.34207239867897523394499674128, −5.08259686021583915230003079693, −3.85899140753777169047869265673, −3.04530881435416008403882118582, −1.95549879912636043081419662789,
1.95549879912636043081419662789, 3.04530881435416008403882118582, 3.85899140753777169047869265673, 5.08259686021583915230003079693, 7.34207239867897523394499674128, 7.80978734107613129448963290835, 10.08078257358470524399030262779, 11.90628051531345037598872307487, 12.68917543023594488072804297644, 13.98937788657479700730583424128
