| L(s) = 1 | + 9i·3-s − 21.1i·5-s + 241.·7-s − 81·9-s + 641. i·11-s − 316. i·13-s + 190.·15-s − 901.·17-s + 2.16e3i·19-s + 2.17e3i·21-s − 397.·23-s + 2.67e3·25-s − 729i·27-s − 5.17e3i·29-s + 2.39e3·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.377i·5-s + 1.86·7-s − 0.333·9-s + 1.59i·11-s − 0.519i·13-s + 0.218·15-s − 0.756·17-s + 1.37i·19-s + 1.07i·21-s − 0.156·23-s + 0.857·25-s − 0.192i·27-s − 1.14i·29-s + 0.446·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.392258049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.392258049\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9iT \) |
| good | 5 | \( 1 + 21.1iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 241.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 641. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 316. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 901.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.16e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 397.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.17e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 2.39e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.35e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.69e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 2.26e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 7.71e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.86e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.07e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.59e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 6.49e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.79e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.07e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.59e4T + 8.58e9T^{2} \) |
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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
−10.56529930536390442853266416126, −10.04796219880112800636581879969, −8.768528204633877625467470116621, −8.113877595315779888750208371220, −7.18880400886525177031291320522, −5.63752594572619486195491275260, −4.72192665039343375291689531522, −4.18676849993654726697773853355, −2.31512082627375235329583057311, −1.29630723101957361365019757221,
0.62309427377989873352467210785, 1.77339007071295132442182071968, 2.94029617897843226032839702022, 4.47408677364286972414757351082, 5.42467678267272954468688015392, 6.59697542534670208454272088565, 7.48536962769780489119335716529, 8.523035768219967460398686886225, 8.941697667429781039750273224370, 10.90758076914406928350189801504
