| L(s) = 1 | + 1.93e3·2-s + 5.90e4·3-s + 1.65e6·4-s + 2.22e7·5-s + 1.14e8·6-s + 4.78e7·7-s − 8.50e8·8-s + 3.48e9·9-s + 4.31e10·10-s + 1.60e11·11-s + 9.79e10·12-s − 7.86e11·13-s + 9.26e10·14-s + 1.31e12·15-s − 5.12e12·16-s − 2.97e12·17-s + 6.75e12·18-s − 2.99e13·19-s + 3.69e13·20-s + 2.82e12·21-s + 3.10e14·22-s − 1.91e14·23-s − 5.01e13·24-s + 1.93e13·25-s − 1.52e15·26-s + 2.05e14·27-s + 7.93e13·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.577·3-s + 0.790·4-s + 1.02·5-s + 0.772·6-s + 0.0639·7-s − 0.279·8-s + 0.333·9-s + 1.36·10-s + 1.86·11-s + 0.456·12-s − 1.58·13-s + 0.0856·14-s + 0.588·15-s − 1.16·16-s − 0.357·17-s + 0.446·18-s − 1.12·19-s + 0.806·20-s + 0.0369·21-s + 2.49·22-s − 0.963·23-s − 0.161·24-s + 0.0405·25-s − 2.11·26-s + 0.192·27-s + 0.0506·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(4.125746030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.125746030\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| good | 2 | \( 1 - 1.93e3T + 2.09e6T^{2} \) |
| 5 | \( 1 - 2.22e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 4.78e7T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.60e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.86e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 2.97e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.99e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.91e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 9.68e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 2.80e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 3.05e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 2.22e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.63e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 4.08e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 4.34e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.14e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 1.98e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.36e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 7.35e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 6.81e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 2.12e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.10e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 7.67e18T + 8.65e40T^{2} \) |
| 97 | \( 1 - 4.63e20T + 5.27e41T^{2} \) |
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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
−21.49544547100370862712309988850, −19.75672159148241859643238816434, −17.32901364553384648815509377694, −14.78307903655031106595731429575, −13.89971987886233398034241738487, −12.26122727322474893680547678385, −9.415981201009002988706469216919, −6.34537009998925288438717855760, −4.31001648806194752940832954182, −2.25684997475839112161469020857,
2.25684997475839112161469020857, 4.31001648806194752940832954182, 6.34537009998925288438717855760, 9.415981201009002988706469216919, 12.26122727322474893680547678385, 13.89971987886233398034241738487, 14.78307903655031106595731429575, 17.32901364553384648815509377694, 19.75672159148241859643238816434, 21.49544547100370862712309988850
