| L(s) = 1 | + 12.7i·3-s + 68.7i·7-s + 80.8·9-s − 327.·11-s − 719. i·13-s + 379. i·17-s + 1.02e3·19-s − 875.·21-s + 779. i·23-s + 4.12e3i·27-s − 1.39e3·29-s − 2.74e3·31-s − 4.16e3i·33-s − 1.26e4i·37-s + 9.15e3·39-s + ⋯ |
| L(s) = 1 | + 0.816i·3-s + 0.530i·7-s + 0.332·9-s − 0.815·11-s − 1.18i·13-s + 0.318i·17-s + 0.654·19-s − 0.433·21-s + 0.307i·23-s + 1.08i·27-s − 0.307·29-s − 0.512·31-s − 0.666i·33-s − 1.51i·37-s + 0.964·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.079440601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.079440601\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 12.7iT - 243T^{2} \) |
| 7 | \( 1 - 68.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 327.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 719. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 379. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.02e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 779. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.74e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.26e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 8.21e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.25e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.73e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.40e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.57e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.20e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 9.00e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.58e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.31e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.45e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.40e4iT - 8.58e9T^{2} \) |
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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
−9.610912612106164403225658424619, −8.893699621869046956671135735884, −7.87189789995515546640176355682, −7.15768061250115474569281528947, −5.62216023215131731629791932722, −5.34842246976781118704632951392, −4.10016567179067524944394990387, −3.22244755165714203620211008301, −2.11016287803006938320477868849, −0.59645907838249661108791674051,
0.73854789005127448723186976423, 1.68672929195459295521547604151, 2.75200125425295286268601844876, 4.06453314555994109258203259378, 4.95032867504483334731387711854, 6.15702938983042087058096311869, 7.02404547117766756135339561110, 7.54191633915585877993845111349, 8.440010889286992075041128644464, 9.539799019028534123720417502021
