| L(s) = 1 | + 1.28·2-s − 0.347·4-s + 0.446·5-s − 3.53·7-s − 3.01·8-s + 0.573·10-s − 2.78·11-s + 3.29·13-s − 4.54·14-s − 3.18·16-s − 7.03·17-s − 5.18·19-s − 0.155·20-s − 3.57·22-s + 7.27·23-s − 4.80·25-s + 4.23·26-s + 1.22·28-s + 3.61·29-s − 1.93·31-s + 1.94·32-s − 9.04·34-s − 1.57·35-s − 3.22·37-s − 6.66·38-s − 1.34·40-s − 4.86·41-s + ⋯ |
| L(s) = 1 | + 0.909·2-s − 0.173·4-s + 0.199·5-s − 1.33·7-s − 1.06·8-s + 0.181·10-s − 0.838·11-s + 0.912·13-s − 1.21·14-s − 0.796·16-s − 1.70·17-s − 1.18·19-s − 0.0346·20-s − 0.761·22-s + 1.51·23-s − 0.960·25-s + 0.829·26-s + 0.231·28-s + 0.672·29-s − 0.347·31-s + 0.343·32-s − 1.55·34-s − 0.266·35-s − 0.530·37-s − 1.08·38-s − 0.213·40-s − 0.759·41-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 | \( 1 - 1.28T + 2T^{2} \) |
| 5 | \( 1 - 0.446T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 + 2.78T + 11T^{2} \) |
| 13 | \( 1 - 3.29T + 13T^{2} \) |
| 17 | \( 1 + 7.03T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 - 7.27T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 + 1.93T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 41 | \( 1 + 4.86T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 - 3.01T + 47T^{2} \) |
| 53 | \( 1 - 8.77T + 53T^{2} \) |
| 59 | \( 1 - 2.96T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 + 9.43T + 67T^{2} \) |
| 71 | \( 1 - 5.30T + 71T^{2} \) |
| 73 | \( 1 + 1.55T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−10.03514844085446347833658885895, −8.963442964275535850844931925271, −8.562559186830470122836872968159, −6.89200209633516030313286753862, −6.33545195003618678078498488534, −5.42491078164367133477291044600, −4.38400877958290081853309367109, −3.46729562625808468602228415298, −2.47101529622553094338710593714, 0,
2.47101529622553094338710593714, 3.46729562625808468602228415298, 4.38400877958290081853309367109, 5.42491078164367133477291044600, 6.33545195003618678078498488534, 6.89200209633516030313286753862, 8.562559186830470122836872968159, 8.963442964275535850844931925271, 10.03514844085446347833658885895
