| L(s) = 1 | − 21.9·2-s + 310.·3-s − 1.56e3·4-s − 886.·5-s − 6.82e3·6-s + 4.42e4·7-s + 7.94e4·8-s − 8.08e4·9-s + 1.94e4·10-s − 4.80e5·11-s − 4.85e5·12-s + 7.29e5·13-s − 9.73e5·14-s − 2.75e5·15-s + 1.45e6·16-s + 6.80e6·17-s + 1.77e6·18-s − 1.56e7·19-s + 1.38e6·20-s + 1.37e7·21-s + 1.05e7·22-s − 3.18e7·23-s + 2.46e7·24-s − 4.80e7·25-s − 1.60e7·26-s − 8.00e7·27-s − 6.92e7·28-s + ⋯ |
| L(s) = 1 | − 0.486·2-s + 0.737·3-s − 0.763·4-s − 0.126·5-s − 0.358·6-s + 0.995·7-s + 0.857·8-s − 0.456·9-s + 0.0616·10-s − 0.899·11-s − 0.563·12-s + 0.544·13-s − 0.483·14-s − 0.0935·15-s + 0.346·16-s + 1.16·17-s + 0.221·18-s − 1.44·19-s + 0.0968·20-s + 0.734·21-s + 0.437·22-s − 1.03·23-s + 0.632·24-s − 0.983·25-s − 0.264·26-s − 1.07·27-s − 0.760·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 2.05e7T \) |
| good | 2 | \( 1 + 21.9T + 2.04e3T^{2} \) |
| 3 | \( 1 - 310.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 886.T + 4.88e7T^{2} \) |
| 7 | \( 1 - 4.42e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 4.80e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 7.29e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.80e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.56e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.18e7T + 9.52e14T^{2} \) |
| 31 | \( 1 + 3.08e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.28e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.00e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 3.53e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.87e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.08e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 9.69e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.73e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.53e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.51e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.05e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.27e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.37e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.92e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.24e11T + 7.15e21T^{2} \) |
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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
−14.09702107604542950663083764859, −12.94425402563309952350110118709, −11.17640987620082984203459106503, −9.841664804433024072473183567546, −8.355183942466229255976878854326, −7.972995242018431303775494057506, −5.40959466469036278463281763038, −3.81557952704054681826734558961, −1.87420858397928547291201097065, 0,
1.87420858397928547291201097065, 3.81557952704054681826734558961, 5.40959466469036278463281763038, 7.972995242018431303775494057506, 8.355183942466229255976878854326, 9.841664804433024072473183567546, 11.17640987620082984203459106503, 12.94425402563309952350110118709, 14.09702107604542950663083764859
