L(s) = 1 | + 3.25i·2-s − 6.59·4-s − 1.59i·5-s − 10.8·7-s − 8.43i·8-s + 5.18·10-s − 9.23·13-s − 35.4i·14-s + 1.07·16-s − 4.81i·17-s − 25.0·19-s + 10.4i·20-s + 36.6i·23-s + 22.4·25-s − 30.0i·26-s + ⋯ |
L(s) = 1 | + 1.62i·2-s − 1.64·4-s − 0.318i·5-s − 1.55·7-s − 1.05i·8-s + 0.518·10-s − 0.710·13-s − 2.52i·14-s + 0.0670·16-s − 0.283i·17-s − 1.32·19-s + 0.524i·20-s + 1.59i·23-s + 0.898·25-s − 1.15i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8443955978\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8443955978\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 3.25iT - 4T^{2} \) |
| 5 | \( 1 + 1.59iT - 25T^{2} \) |
| 7 | \( 1 + 10.8T + 49T^{2} \) |
| 13 | \( 1 + 9.23T + 169T^{2} \) |
| 17 | \( 1 + 4.81iT - 289T^{2} \) |
| 19 | \( 1 + 25.0T + 361T^{2} \) |
| 23 | \( 1 - 36.6iT - 529T^{2} \) |
| 29 | \( 1 + 24.4iT - 841T^{2} \) |
| 31 | \( 1 - 58.1T + 961T^{2} \) |
| 37 | \( 1 - 12.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 12.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 19.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 64.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 49.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 24.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 44.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 33.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 74.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 41.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 63.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 42.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 108.T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.619017310511091967231556582665, −8.784903476085112321676904344106, −8.043642085782911385127994479551, −7.06075753118278227597360845741, −6.55294370357242928054115005812, −5.79413857802294381438543149905, −4.88508738551296979801223220751, −3.91462862481012430214863066455, −2.60353165038020520591164055045, −0.37845016224431688730461973825,
0.76545220777997325938297694865, 2.45553720994248026404350678925, 2.88215557222561923918819593118, 3.96802479603320235018505372155, 4.77476243558394662224693316771, 6.35096796923009763499933022601, 6.76865332468764086106218773958, 8.262420288659151999820215660898, 9.141386446265714646494505356279, 9.768697238329096566293515200970