L(s) = 1 | + 2.11·2-s + 2.48·4-s + 2.68·5-s + 0.972·7-s + 1.01·8-s + 5.67·10-s + 0.316·11-s − 1.51·13-s + 2.05·14-s − 2.80·16-s − 1.17·17-s + 6.22·19-s + 6.65·20-s + 0.670·22-s − 2.16·23-s + 2.19·25-s − 3.20·26-s + 2.41·28-s + 4.40·29-s − 8.67·31-s − 7.97·32-s − 2.48·34-s + 2.60·35-s − 4.46·37-s + 13.1·38-s + 2.72·40-s − 5.84·41-s + ⋯ |
L(s) = 1 | + 1.49·2-s + 1.24·4-s + 1.19·5-s + 0.367·7-s + 0.359·8-s + 1.79·10-s + 0.0955·11-s − 0.419·13-s + 0.550·14-s − 0.702·16-s − 0.284·17-s + 1.42·19-s + 1.48·20-s + 0.143·22-s − 0.450·23-s + 0.439·25-s − 0.628·26-s + 0.455·28-s + 0.817·29-s − 1.55·31-s − 1.41·32-s − 0.426·34-s + 0.440·35-s − 0.734·37-s + 2.13·38-s + 0.431·40-s − 0.912·41-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
\(L(1)\) |
\(\approx\) |
\(3.949542196\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.949542196\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.11T + 2T^{2} \) |
| 5 | \( 1 - 2.68T + 5T^{2} \) |
| 7 | \( 1 - 0.972T + 7T^{2} \) |
| 11 | \( 1 - 0.316T + 11T^{2} \) |
| 13 | \( 1 + 1.51T + 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 - 6.22T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 4.40T + 29T^{2} \) |
| 31 | \( 1 + 8.67T + 31T^{2} \) |
| 37 | \( 1 + 4.46T + 37T^{2} \) |
| 41 | \( 1 + 5.84T + 41T^{2} \) |
| 43 | \( 1 - 5.59T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 - 1.01T + 61T^{2} \) |
| 67 | \( 1 - 0.856T + 67T^{2} \) |
| 71 | \( 1 - 9.59T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 4.68T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−10.50163698683390368747460255548, −9.637561137205647863350398401625, −8.813836943935062289590415105321, −7.42714740929352880147791074673, −6.56061272223061047440280848627, −5.56285365207551545232442198576, −5.18499163173812978906141241042, −4.01869655240563557747779861066, −2.87142918998076205608054637174, −1.81368532216312820667387991120,
1.81368532216312820667387991120, 2.87142918998076205608054637174, 4.01869655240563557747779861066, 5.18499163173812978906141241042, 5.56285365207551545232442198576, 6.56061272223061047440280848627, 7.42714740929352880147791074673, 8.813836943935062289590415105321, 9.637561137205647863350398401625, 10.50163698683390368747460255548