| L(s) = 1 | + 1.71e3·2-s + 8.50e5·4-s − 9.76e6·5-s − 6.24e8·7-s − 2.14e9·8-s − 1.67e10·10-s − 1.47e11·11-s − 5.50e11·13-s − 1.07e12·14-s − 5.45e12·16-s + 4.04e12·17-s + 1.22e12·19-s − 8.30e12·20-s − 2.53e14·22-s + 2.50e14·23-s + 9.53e13·25-s − 9.45e14·26-s − 5.31e14·28-s + 3.49e15·29-s − 2.93e15·31-s − 4.88e15·32-s + 6.95e15·34-s + 6.09e15·35-s + 4.06e16·37-s + 2.10e15·38-s + 2.09e16·40-s − 1.45e17·41-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.405·4-s − 0.447·5-s − 0.835·7-s − 0.704·8-s − 0.530·10-s − 1.71·11-s − 1.10·13-s − 0.990·14-s − 1.24·16-s + 0.487·17-s + 0.0459·19-s − 0.181·20-s − 2.03·22-s + 1.26·23-s + 0.199·25-s − 1.31·26-s − 0.338·28-s + 1.54·29-s − 0.642·31-s − 0.766·32-s + 0.577·34-s + 0.373·35-s + 1.38·37-s + 0.0544·38-s + 0.315·40-s − 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.491423126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491423126\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 9.76e6T \) |
| good | 2 | \( 1 - 1.71e3T + 2.09e6T^{2} \) |
| 7 | \( 1 + 6.24e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.47e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 5.50e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 4.04e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.22e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.50e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.49e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.93e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.06e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.45e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 8.05e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.95e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.24e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 7.20e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 4.77e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 7.70e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.63e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.29e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 9.55e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.20e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.15e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 2.46e20T + 5.27e41T^{2} \) |
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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
−12.08752824832366251081019899665, −10.62162541619406821808652968527, −9.434043916472290209117824583624, −7.925661808048971603484989154936, −6.71548747807764158673202294660, −5.37802426682904202908343187039, −4.64948348735575554198424198864, −3.20942321412360675521358032589, −2.63281472626392836416860681181, −0.44074888670186003444212388732,
0.44074888670186003444212388732, 2.63281472626392836416860681181, 3.20942321412360675521358032589, 4.64948348735575554198424198864, 5.37802426682904202908343187039, 6.71548747807764158673202294660, 7.925661808048971603484989154936, 9.434043916472290209117824583624, 10.62162541619406821808652968527, 12.08752824832366251081019899665
