| L(s) = 1 | + 16.7i·3-s + 10.1·5-s + 80.4i·7-s − 198.·9-s + 21.6i·11-s + 179.·13-s + 170. i·15-s + 148.·17-s − 322. i·19-s − 1.34e3·21-s + 327. i·23-s − 521.·25-s − 1.96e3i·27-s − 699.·29-s − 1.04e3i·31-s + ⋯ |
| L(s) = 1 | + 1.85i·3-s + 0.407·5-s + 1.64i·7-s − 2.44·9-s + 0.178i·11-s + 1.05·13-s + 0.756i·15-s + 0.513·17-s − 0.893i·19-s − 3.04·21-s + 0.618i·23-s − 0.833·25-s − 2.69i·27-s − 0.831·29-s − 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.60647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60647i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 16.7iT - 81T^{2} \) |
| 5 | \( 1 - 10.1T + 625T^{2} \) |
| 7 | \( 1 - 80.4iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 21.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 179.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 148.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 322. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 327. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 699.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.04e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.14e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 792.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 231. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.96e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.67e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.05e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.91e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.62e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.92e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.35e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 5.65e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.22e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.19e3T + 8.85e7T^{2} \) |
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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
−13.23755853945083770773925803177, −11.71755708250580511448813825011, −11.09574356953270856541948750722, −9.755269348423229436417637774256, −9.260919351651311050253022415148, −8.319244739352527896329891437257, −5.93253983527424764512135668276, −5.34920600801402458498167291425, −3.92056999575437876520277425544, −2.57707470582259338561721827577,
0.68804769711466102980485309064, 1.72185977882040235572769748051, 3.59971977022250947087668379893, 5.77370622900778581994207751045, 6.74617358125477522795006802801, 7.61301562831891031648089394798, 8.507897715840279079768455323286, 10.21779607638736286245292528163, 11.24932628161721455559187408341, 12.33966934443258846980974233327
