| L(s) = 1 | − 1.93e3·2-s + 1.65e6·4-s − 2.22e7·5-s + 4.78e7·7-s + 8.50e8·8-s + 4.31e10·10-s − 1.60e11·11-s − 7.86e11·13-s − 9.26e10·14-s − 5.12e12·16-s + 2.97e12·17-s − 2.99e13·19-s − 3.69e13·20-s + 3.10e14·22-s + 1.91e14·23-s + 1.93e13·25-s + 1.52e15·26-s + 7.93e13·28-s − 9.68e14·29-s + 2.80e15·31-s + 8.15e15·32-s − 5.76e15·34-s − 1.06e15·35-s + 3.05e16·37-s + 5.80e16·38-s − 1.89e16·40-s + 2.22e16·41-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.790·4-s − 1.02·5-s + 0.0639·7-s + 0.279·8-s + 1.36·10-s − 1.86·11-s − 1.58·13-s − 0.0856·14-s − 1.16·16-s + 0.357·17-s − 1.12·19-s − 0.806·20-s + 2.49·22-s + 0.963·23-s + 0.0405·25-s + 2.11·26-s + 0.0506·28-s − 0.427·29-s + 0.614·31-s + 1.27·32-s − 0.478·34-s − 0.0652·35-s + 1.04·37-s + 1.49·38-s − 0.285·40-s + 0.259·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.2717356895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2717356895\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 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
−16.28840568283866407544562494979, −15.05653681714102024596849310279, −12.81626208261011362283744084379, −11.09432143062809884988752872425, −9.881748268042164303783479361149, −8.174762702916693838669892475364, −7.40498662748837978922671546410, −4.75658686997557929485351830652, −2.47909800498750747947282194614, −0.39441388374342991176490321881,
0.39441388374342991176490321881, 2.47909800498750747947282194614, 4.75658686997557929485351830652, 7.40498662748837978922671546410, 8.174762702916693838669892475364, 9.881748268042164303783479361149, 11.09432143062809884988752872425, 12.81626208261011362283744084379, 15.05653681714102024596849310279, 16.28840568283866407544562494979
