L(s) = 1 | + 2.23i·2-s − 3.00·4-s − 3.87·5-s + (−2 − 1.73i)7-s − 2.23i·8-s − 8.66i·10-s + 2.23i·11-s + 3.46i·13-s + (3.87 − 4.47i)14-s − 0.999·16-s + 5.19i·19-s + 11.6·20-s − 5.00·22-s + 2.23i·23-s + 10.0·25-s − 7.74·26-s + ⋯ |
L(s) = 1 | + 1.58i·2-s − 1.50·4-s − 1.73·5-s + (−0.755 − 0.654i)7-s − 0.790i·8-s − 2.73i·10-s + 0.674i·11-s + 0.960i·13-s + (1.03 − 1.19i)14-s − 0.249·16-s + 1.19i·19-s + 2.59·20-s − 1.06·22-s + 0.466i·23-s + 2.00·25-s − 1.51·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.174168 - 0.381237i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.174168 - 0.381237i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 2 | \( 1 - 2.23iT - 2T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 11 | \( 1 - 2.23iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 2.23iT - 23T^{2} \) |
| 29 | \( 1 + 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 3.87T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 7.74T + 47T^{2} \) |
| 53 | \( 1 - 8.94iT - 53T^{2} \) |
| 59 | \( 1 - 7.74T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 11.1iT - 71T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 7.74T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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
−13.44276526376598835209073479918, −12.28208574294856396926463025151, −11.40809757460803020987198133840, −9.971268747421616652116100862803, −8.771312455738900040001387011517, −7.70462974402408316207458772645, −7.23712989446454673679922460951, −6.22291742876089152161715416877, −4.57893158611956600816912200405, −3.78254987793684010819448330215,
0.37132668927245395839836744855, 2.92167662656357775997283223476, 3.58618483410507751004618352609, 4.96082542505710999669918964658, 6.83104194364633565449063006377, 8.269443455592867292505575837509, 9.034931360600023406680396821073, 10.32785135302290874689779786250, 11.18015667075553484160437546147, 11.82816246192793014299284808084