| L(s) = 1 | − 0.777·2-s − 1.39·4-s − 2.37·5-s − 2.50·7-s + 2.64·8-s + 1.84·10-s − 3.14·11-s + 1.33·13-s + 1.94·14-s + 0.736·16-s − 6.27·17-s + 8.06·19-s + 3.31·20-s + 2.44·22-s − 4.05·23-s + 0.647·25-s − 1.03·26-s + 3.48·28-s + 9.28·29-s + 2.83·31-s − 5.85·32-s + 4.87·34-s + 5.94·35-s + 5.53·37-s − 6.27·38-s − 6.27·40-s + 7.10·41-s + ⋯ |
| L(s) = 1 | − 0.549·2-s − 0.697·4-s − 1.06·5-s − 0.945·7-s + 0.933·8-s + 0.584·10-s − 0.947·11-s + 0.370·13-s + 0.519·14-s + 0.184·16-s − 1.52·17-s + 1.85·19-s + 0.741·20-s + 0.520·22-s − 0.845·23-s + 0.129·25-s − 0.203·26-s + 0.659·28-s + 1.72·29-s + 0.508·31-s − 1.03·32-s + 0.836·34-s + 1.00·35-s + 0.909·37-s − 1.01·38-s − 0.992·40-s + 1.10·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5018737572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5018737572\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.777T + 2T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 + 6.27T + 17T^{2} \) |
| 19 | \( 1 - 8.06T + 19T^{2} \) |
| 23 | \( 1 + 4.05T + 23T^{2} \) |
| 29 | \( 1 - 9.28T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 - 5.53T + 37T^{2} \) |
| 41 | \( 1 - 7.10T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 4.61T + 47T^{2} \) |
| 53 | \( 1 + 0.135T + 53T^{2} \) |
| 59 | \( 1 - 3.99T + 59T^{2} \) |
| 61 | \( 1 - 0.341T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 4.08T + 79T^{2} \) |
| 83 | \( 1 + 0.913T + 83T^{2} \) |
| 89 | \( 1 - 3.72T + 89T^{2} \) |
| 97 | \( 1 + 5.99T + 97T^{2} \) |
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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
−10.20222123188108201090754213599, −9.552175772106926569770528124177, −8.634623227497148766485338745519, −7.922767231031994151139273256571, −7.20315680453053910947914496554, −6.00811829758901071870828606034, −4.76125146347665162354437039961, −3.94904872901037551523882451512, −2.80807081801876942619912520470, −0.62934138515479486615391030088,
0.62934138515479486615391030088, 2.80807081801876942619912520470, 3.94904872901037551523882451512, 4.76125146347665162354437039961, 6.00811829758901071870828606034, 7.20315680453053910947914496554, 7.922767231031994151139273256571, 8.634623227497148766485338745519, 9.552175772106926569770528124177, 10.20222123188108201090754213599
