| L(s) = 1 | + 1.77e3·2-s + 1.05e6·4-s + 3.60e7·5-s − 2.31e8·7-s − 1.85e9·8-s + 6.40e10·10-s − 3.66e10·11-s − 2.05e11·13-s − 4.10e11·14-s − 5.49e12·16-s + 4.27e12·17-s + 3.46e13·19-s + 3.80e13·20-s − 6.50e13·22-s − 2.66e14·23-s + 8.22e14·25-s − 3.64e14·26-s − 2.43e14·28-s − 3.18e15·29-s − 3.91e15·31-s − 5.88e15·32-s + 7.59e15·34-s − 8.33e15·35-s + 1.55e16·37-s + 6.15e16·38-s − 6.66e16·40-s + 1.45e17·41-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.503·4-s + 1.65·5-s − 0.309·7-s − 0.609·8-s + 2.02·10-s − 0.426·11-s − 0.413·13-s − 0.379·14-s − 1.24·16-s + 0.514·17-s + 1.29·19-s + 0.830·20-s − 0.522·22-s − 1.34·23-s + 1.72·25-s − 0.507·26-s − 0.155·28-s − 1.40·29-s − 0.857·31-s − 0.923·32-s + 0.630·34-s − 0.510·35-s + 0.530·37-s + 1.58·38-s − 1.00·40-s + 1.69·41-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 | \( 1 - 1.77e3T + 2.09e6T^{2} \) |
| 5 | \( 1 - 3.60e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 2.31e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 3.66e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 2.05e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 4.27e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.46e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.66e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.18e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 3.91e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.55e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.45e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 8.81e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 5.74e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.50e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 6.32e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 6.64e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.15e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.65e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.51e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.10e20T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.54e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.58e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 3.44e20T + 5.27e41T^{2} \) |
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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
−9.755102338238550187524276408248, −9.385605260104817648812161835150, −7.60423733515818238535010334612, −6.16490510869739423922933126040, −5.68188015425883563115239633041, −4.81096446411564654256963426704, −3.44760405115292549477184643069, −2.54080915967457218600045043680, −1.56517441648243970620988824663, 0,
1.56517441648243970620988824663, 2.54080915967457218600045043680, 3.44760405115292549477184643069, 4.81096446411564654256963426704, 5.68188015425883563115239633041, 6.16490510869739423922933126040, 7.60423733515818238535010334612, 9.385605260104817648812161835150, 9.755102338238550187524276408248
