| L(s) = 1 | + 11.1·2-s + 93.1·4-s − 62.3·5-s + 684.·8-s − 697.·10-s + 173.·11-s − 348.·13-s + 4.67e3·16-s + 999.·17-s + 2.04e3·19-s − 5.80e3·20-s + 1.94e3·22-s + 1.94e3·23-s + 758.·25-s − 3.89e3·26-s − 738.·29-s + 2.45e3·31-s + 3.04e4·32-s + 1.11e4·34-s + 8.78e3·37-s + 2.28e4·38-s − 4.26e4·40-s + 1.76e4·41-s − 1.03e4·43-s + 1.61e4·44-s + 2.18e4·46-s − 75.0·47-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 2.91·4-s − 1.11·5-s + 3.78·8-s − 2.20·10-s + 0.432·11-s − 0.571·13-s + 4.56·16-s + 0.838·17-s + 1.29·19-s − 3.24·20-s + 0.855·22-s + 0.768·23-s + 0.242·25-s − 1.13·26-s − 0.162·29-s + 0.459·31-s + 5.24·32-s + 1.65·34-s + 1.05·37-s + 2.56·38-s − 4.21·40-s + 1.63·41-s − 0.850·43-s + 1.25·44-s + 1.51·46-s − 0.00495·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(7.790190681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.790190681\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 11.1T + 32T^{2} \) |
| 5 | \( 1 + 62.3T + 3.12e3T^{2} \) |
| 11 | \( 1 - 173.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 348.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 999.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.94e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 738.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.45e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.78e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.76e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 75.0T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.77e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.45e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.61e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.63e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.14e4T + 8.58e9T^{2} \) |
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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
−10.89167379805071769818672790905, −9.672063406068242244588554390863, −7.83334091331030743594535406419, −7.41085673476964567656704530162, −6.34430000620857544998002799065, −5.29590148362260197864130931573, −4.45979630464461278771102329575, −3.55797264822193827476346445515, −2.77397070917829880696809077148, −1.17657452478092444167549550723,
1.17657452478092444167549550723, 2.77397070917829880696809077148, 3.55797264822193827476346445515, 4.45979630464461278771102329575, 5.29590148362260197864130931573, 6.34430000620857544998002799065, 7.41085673476964567656704530162, 7.83334091331030743594535406419, 9.672063406068242244588554390863, 10.89167379805071769818672790905
