| L(s) = 1 | − 8.44·2-s + 39.2·4-s + 25.1·5-s − 71.3·7-s − 61.2·8-s − 211.·10-s − 509.·11-s − 341.·13-s + 602.·14-s − 738.·16-s − 1.62e3·17-s − 1.23e3·19-s + 985.·20-s + 4.29e3·22-s − 3.18e3·23-s − 2.49e3·25-s + 2.88e3·26-s − 2.80e3·28-s − 5.53e3·29-s − 2.85e3·31-s + 8.19e3·32-s + 1.37e4·34-s − 1.79e3·35-s − 1.07e4·37-s + 1.04e4·38-s − 1.53e3·40-s + 1.89e4·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.22·4-s + 0.449·5-s − 0.550·7-s − 0.338·8-s − 0.670·10-s − 1.26·11-s − 0.560·13-s + 0.821·14-s − 0.721·16-s − 1.36·17-s − 0.783·19-s + 0.551·20-s + 1.89·22-s − 1.25·23-s − 0.798·25-s + 0.836·26-s − 0.675·28-s − 1.22·29-s − 0.534·31-s + 1.41·32-s + 2.03·34-s − 0.247·35-s − 1.28·37-s + 1.16·38-s − 0.152·40-s + 1.75·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.03137986154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03137986154\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 8.44T + 32T^{2} \) |
| 5 | \( 1 - 25.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 71.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 509.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 341.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.62e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.23e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.53e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.85e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.89e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.39e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 353.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.02e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.57e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.57e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.61e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.19e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.05e4T + 8.58e9T^{2} \) |
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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
−9.629434288744881756444631503696, −8.877841443718384580121751590036, −8.048899826733830438375637560733, −7.28335822776090441544199952957, −6.39807560061486073800206366572, −5.35710653318101575603028258495, −4.06896513612853129681398157650, −2.43827530037269697670704209668, −1.89355254774637462817716787471, −0.096393495900243005539552389063,
0.096393495900243005539552389063, 1.89355254774637462817716787471, 2.43827530037269697670704209668, 4.06896513612853129681398157650, 5.35710653318101575603028258495, 6.39807560061486073800206366572, 7.28335822776090441544199952957, 8.048899826733830438375637560733, 8.877841443718384580121751590036, 9.629434288744881756444631503696
