| L(s) = 1 | + 11.8·2-s − 1.90e3·4-s + 4.02e3·5-s − 6.97e4·7-s − 4.67e4·8-s + 4.76e4·10-s − 8.37e5·11-s + 6.28e4·13-s − 8.24e5·14-s + 3.35e6·16-s − 6.17e6·17-s + 8.86e6·19-s − 7.68e6·20-s − 9.89e6·22-s + 2.86e7·23-s − 3.25e7·25-s + 7.43e5·26-s + 1.33e8·28-s − 1.35e8·29-s − 1.27e8·31-s + 1.35e8·32-s − 7.30e7·34-s − 2.80e8·35-s + 7.85e7·37-s + 1.04e8·38-s − 1.88e8·40-s + 5.90e8·41-s + ⋯ |
| L(s) = 1 | + 0.261·2-s − 0.931·4-s + 0.576·5-s − 1.56·7-s − 0.504·8-s + 0.150·10-s − 1.56·11-s + 0.0469·13-s − 0.409·14-s + 0.799·16-s − 1.05·17-s + 0.821·19-s − 0.537·20-s − 0.409·22-s + 0.927·23-s − 0.667·25-s + 0.0122·26-s + 1.46·28-s − 1.22·29-s − 0.801·31-s + 0.713·32-s − 0.275·34-s − 0.903·35-s + 0.186·37-s + 0.214·38-s − 0.291·40-s + 0.795·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.8115886005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8115886005\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 11.8T + 2.04e3T^{2} \) |
| 5 | \( 1 - 4.02e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.97e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.37e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 6.28e4T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.17e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.86e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.86e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.35e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.27e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 7.85e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.90e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 6.26e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.16e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.08e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 8.14e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 2.86e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 4.10e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.70e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.87e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.29e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.78e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.82e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 4.73e10T + 7.15e21T^{2} \) |
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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
−12.65033985621412712240524641989, −10.81851777760997201396224783417, −9.685353883436031190482859001158, −9.104213636653052482515515709774, −7.52552920351709525854047858893, −6.05846526315535521031533616088, −5.16204563980872945477557359443, −3.64532950967753667650455776173, −2.52327026189644041413347532007, −0.44137780201943631463128426628,
0.44137780201943631463128426628, 2.52327026189644041413347532007, 3.64532950967753667650455776173, 5.16204563980872945477557359443, 6.05846526315535521031533616088, 7.52552920351709525854047858893, 9.104213636653052482515515709774, 9.685353883436031190482859001158, 10.81851777760997201396224783417, 12.65033985621412712240524641989
