| L(s) = 1 | + 6.06·2-s − 282.·3-s − 2.01e3·4-s − 8.65e3·5-s − 1.71e3·6-s − 4.26e4·7-s − 2.46e4·8-s − 9.73e4·9-s − 5.24e4·10-s − 8.47e4·11-s + 5.68e5·12-s − 1.84e5·13-s − 2.58e5·14-s + 2.44e6·15-s + 3.96e6·16-s − 3.55e6·17-s − 5.90e5·18-s − 5.58e6·19-s + 1.74e7·20-s + 1.20e7·21-s − 5.13e5·22-s − 9.75e6·23-s + 6.95e6·24-s + 2.61e7·25-s − 1.11e6·26-s + 7.75e7·27-s + 8.56e7·28-s + ⋯ |
| L(s) = 1 | + 0.134·2-s − 0.671·3-s − 0.982·4-s − 1.23·5-s − 0.0899·6-s − 0.958·7-s − 0.265·8-s − 0.549·9-s − 0.166·10-s − 0.158·11-s + 0.659·12-s − 0.137·13-s − 0.128·14-s + 0.831·15-s + 0.946·16-s − 0.607·17-s − 0.0736·18-s − 0.517·19-s + 1.21·20-s + 0.643·21-s − 0.0212·22-s − 0.316·23-s + 0.178·24-s + 0.534·25-s − 0.0184·26-s + 1.03·27-s + 0.940·28-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
| \(L(6)\) |
\(\approx\) |
\(0.1423214167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1423214167\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 6.06T + 2.04e3T^{2} \) |
| 3 | \( 1 + 282.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 8.65e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 4.26e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.47e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.84e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.55e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 5.58e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 9.75e6T + 9.52e14T^{2} \) |
| 31 | \( 1 + 1.20e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.53e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.41e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.80e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.58e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.78e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 4.38e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.17e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 6.71e8T + 1.22e20T^{2} \) |
| 71 | \( 1 - 3.09e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 4.55e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 6.82e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.79e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.74e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 6.18e10T + 7.15e21T^{2} \) |
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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
−14.60534715494674961446296153217, −13.10691200781887890615628423434, −12.15087737325012152133686543642, −10.91679648690429159337150587436, −9.316150514151334928568964382423, −7.985169484557705388581950382347, −6.22176899039568123414543242685, −4.66559604777726811654149729688, −3.36469385193989194082147309328, −0.24466279364403572985299529769,
0.24466279364403572985299529769, 3.36469385193989194082147309328, 4.66559604777726811654149729688, 6.22176899039568123414543242685, 7.985169484557705388581950382347, 9.316150514151334928568964382423, 10.91679648690429159337150587436, 12.15087737325012152133686543642, 13.10691200781887890615628423434, 14.60534715494674961446296153217
