| 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
| \(L(1)\) |
\(\approx\) |
\(4.457404053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.457404053\) |
| \(L(\frac{3}{2})\) |
|
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
−10.83825390502468082769418293347, −9.895134984675393727838785945861, −8.206352075220452793699290250844, −7.56942209708635272007851485562, −6.64584399877314293437244697816, −5.58745861719681841202894793965, −5.01427103632711479399913982634, −3.74249690306772619464437519023, −3.35179486421818005857654391513, −1.81702693168411902282057463504,
1.81702693168411902282057463504, 3.35179486421818005857654391513, 3.74249690306772619464437519023, 5.01427103632711479399913982634, 5.58745861719681841202894793965, 6.64584399877314293437244697816, 7.56942209708635272007851485562, 8.206352075220452793699290250844, 9.895134984675393727838785945861, 10.83825390502468082769418293347
