| L(s) = 1 | + 3.46i·3-s − 20.7·5-s + 40i·7-s + 69·9-s − 197. i·11-s + 256.·13-s − 72i·15-s − 198·17-s + 252. i·19-s − 138.·21-s + 504i·23-s − 193·25-s + 519. i·27-s − 436.·29-s + 1.69e3i·31-s + ⋯ |
| L(s) = 1 | + 0.384i·3-s − 0.831·5-s + 0.816i·7-s + 0.851·9-s − 1.63i·11-s + 1.51·13-s − 0.320i·15-s − 0.685·17-s + 0.700i·19-s − 0.314·21-s + 0.952i·23-s − 0.308·25-s + 0.712i·27-s − 0.518·29-s + 1.76i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.530630254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530630254\) |
| \(L(3)\) |
|
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 | \( 1 - 3.46iT - 81T^{2} \) |
| 5 | \( 1 + 20.7T + 625T^{2} \) |
| 7 | \( 1 - 40iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 197. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 256.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 198T + 8.35e4T^{2} \) |
| 19 | \( 1 - 252. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 504iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 436.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.69e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.75e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 18T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.54e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.44e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.26e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 5.26e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.00e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.80e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 7.84e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.31e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.14e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.09e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.11e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 4.73e3T + 8.85e7T^{2} \) |
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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
−11.24560223933406199740665184647, −11.05021203535761787458929885289, −9.579728315068198943762510850135, −8.622214390808280339897675048098, −7.940643205324387560666761207020, −6.46831833986005163853187082882, −5.52723459967610904934347756967, −4.05386187997231138658127777893, −3.26380151158554959088248643673, −1.26450010582141770921280052349,
0.56866233069926036421326714544, 2.00357239349878028638333552331, 3.97571582305323512110496162168, 4.46371337251759815778079944663, 6.37308532947182396923180753054, 7.25413223164216277314200698313, 7.897039083407760429566505508669, 9.204426143797476456976257600033, 10.27828140687234401346328217294, 11.14496542837179475472815173531
