| L(s) = 1 | − 0.801·2-s − 1.35·4-s − 2.74·5-s + 2.37·7-s + 2.69·8-s + 2.20·10-s − 0.250·11-s + 2.61·13-s − 1.90·14-s + 0.558·16-s − 0.293·17-s − 2.78·19-s + 3.73·20-s + 0.200·22-s − 6.68·23-s + 2.56·25-s − 2.09·26-s − 3.22·28-s − 0.355·29-s + 2.76·31-s − 5.82·32-s + 0.235·34-s − 6.53·35-s − 6.99·37-s + 2.23·38-s − 7.40·40-s − 9.71·41-s + ⋯ |
| L(s) = 1 | − 0.566·2-s − 0.678·4-s − 1.22·5-s + 0.898·7-s + 0.951·8-s + 0.696·10-s − 0.0754·11-s + 0.724·13-s − 0.509·14-s + 0.139·16-s − 0.0711·17-s − 0.638·19-s + 0.834·20-s + 0.0427·22-s − 1.39·23-s + 0.512·25-s − 0.410·26-s − 0.609·28-s − 0.0659·29-s + 0.496·31-s − 1.03·32-s + 0.0403·34-s − 1.10·35-s − 1.14·37-s + 0.362·38-s − 1.17·40-s − 1.51·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.801T + 2T^{2} \) |
| 5 | \( 1 + 2.74T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 0.250T + 11T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 + 0.293T + 17T^{2} \) |
| 19 | \( 1 + 2.78T + 19T^{2} \) |
| 23 | \( 1 + 6.68T + 23T^{2} \) |
| 29 | \( 1 + 0.355T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 + 6.99T + 37T^{2} \) |
| 41 | \( 1 + 9.71T + 41T^{2} \) |
| 43 | \( 1 - 0.260T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 - 5.97T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 1.81T + 67T^{2} \) |
| 71 | \( 1 + 0.370T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 - 0.802T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−10.01339946902994293345861198063, −8.831067808765457271448301803388, −8.182622502921477939529670707860, −7.85287367684203177135464663525, −6.64048781848745687638880781739, −5.23447279546781579907709607211, −4.33997754212319295297559547938, −3.61316423255547458311858525856, −1.64306494038862216364660489403, 0,
1.64306494038862216364660489403, 3.61316423255547458311858525856, 4.33997754212319295297559547938, 5.23447279546781579907709607211, 6.64048781848745687638880781739, 7.85287367684203177135464663525, 8.182622502921477939529670707860, 8.831067808765457271448301803388, 10.01339946902994293345861198063
