L(s) = 1 | − 2.70·2-s + 5.31·4-s + 1.67·5-s + 0.500·7-s − 8.97·8-s − 4.52·10-s + 1.91·11-s + 3.11·13-s − 1.35·14-s + 13.6·16-s − 2.66·17-s + 5.79·19-s + 8.89·20-s − 5.18·22-s + 4.64·23-s − 2.20·25-s − 8.41·26-s + 2.66·28-s + 2.61·29-s − 4.61·31-s − 18.9·32-s + 7.20·34-s + 0.837·35-s − 4.85·37-s − 15.6·38-s − 15.0·40-s + 11.5·41-s + ⋯ |
L(s) = 1 | − 1.91·2-s + 2.65·4-s + 0.747·5-s + 0.189·7-s − 3.17·8-s − 1.43·10-s + 0.578·11-s + 0.863·13-s − 0.361·14-s + 3.40·16-s − 0.646·17-s + 1.32·19-s + 1.98·20-s − 1.10·22-s + 0.968·23-s − 0.440·25-s − 1.65·26-s + 0.503·28-s + 0.485·29-s − 0.828·31-s − 3.34·32-s + 1.23·34-s + 0.141·35-s − 0.798·37-s − 2.54·38-s − 2.37·40-s + 1.80·41-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
\(L(1)\) |
\(\approx\) |
\(0.8248511921\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8248511921\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 7 | \( 1 - 0.500T + 7T^{2} \) |
| 11 | \( 1 - 1.91T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 + 2.66T + 17T^{2} \) |
| 19 | \( 1 - 5.79T + 19T^{2} \) |
| 23 | \( 1 - 4.64T + 23T^{2} \) |
| 29 | \( 1 - 2.61T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 9.00T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 - 6.84T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 - 9.93T + 73T^{2} \) |
| 79 | \( 1 - 5.31T + 79T^{2} \) |
| 83 | \( 1 - 2.72T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 6.88T + 97T^{2} \) |
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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
−10.07292211102659265671465369155, −9.490321889607855106065507122502, −8.837563876924864740450621561048, −8.066940358342097176007126777810, −7.04209747848816271722456565404, −6.40492179560029008432051775714, −5.37843378445568793119819717287, −3.40388547951839954470447772326, −2.08243058422224331483037034773, −1.07623755241472714143105800585,
1.07623755241472714143105800585, 2.08243058422224331483037034773, 3.40388547951839954470447772326, 5.37843378445568793119819717287, 6.40492179560029008432051775714, 7.04209747848816271722456565404, 8.066940358342097176007126777810, 8.837563876924864740450621561048, 9.490321889607855106065507122502, 10.07292211102659265671465369155