| L(s) = 1 | + 11.8·2-s − 2.09e6·4-s + 3.62e7·5-s − 2.43e8·7-s − 4.98e7·8-s + 4.30e8·10-s − 6.05e10·11-s + 5.18e11·13-s − 2.89e9·14-s + 4.39e12·16-s + 6.71e12·17-s − 4.48e13·19-s − 7.60e13·20-s − 7.19e11·22-s − 2.35e14·23-s + 8.37e14·25-s + 6.16e12·26-s + 5.10e14·28-s + 4.37e15·29-s − 3.36e15·31-s + 1.56e14·32-s + 7.98e13·34-s − 8.82e15·35-s − 2.76e16·37-s − 5.33e14·38-s − 1.80e15·40-s + 2.84e16·41-s + ⋯ |
| L(s) = 1 | + 0.00820·2-s − 0.999·4-s + 1.65·5-s − 0.325·7-s − 0.0164·8-s + 0.0136·10-s − 0.704·11-s + 1.04·13-s − 0.00267·14-s + 0.999·16-s + 0.807·17-s − 1.67·19-s − 1.65·20-s − 0.00577·22-s − 1.18·23-s + 1.75·25-s + 0.00855·26-s + 0.325·28-s + 1.92·29-s − 0.737·31-s + 0.0246·32-s + 0.00662·34-s − 0.540·35-s − 0.945·37-s − 0.0137·38-s − 0.0272·40-s + 0.331·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 11.8T + 2.09e6T^{2} \) |
| 5 | \( 1 - 3.62e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 2.43e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 6.05e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 5.18e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 6.71e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.48e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.35e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 4.37e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 3.36e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 2.76e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 2.84e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 9.68e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 7.02e13T + 1.30e35T^{2} \) |
| 53 | \( 1 - 5.90e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.10e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.46e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.96e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 1.57e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.25e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 7.23e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.21e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.75e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.18e21T + 5.27e41T^{2} \) |
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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
−10.09756843441238577838051886190, −8.971075974435935920120254931261, −8.205811777837859257041054774739, −6.40278546126147714735625050361, −5.75116618191378379400884772050, −4.73906960359935046316744804085, −3.46125649936722945924365415792, −2.21335306181674228965337778515, −1.20545986399593910924881034455, 0,
1.20545986399593910924881034455, 2.21335306181674228965337778515, 3.46125649936722945924365415792, 4.73906960359935046316744804085, 5.75116618191378379400884772050, 6.40278546126147714735625050361, 8.205811777837859257041054774739, 8.971075974435935920120254931261, 10.09756843441238577838051886190
