L(s) = 1 | + (1.90 − 0.601i)2-s + (3.27 − 2.29i)4-s − 8.03·5-s + 8.62i·7-s + (4.87 − 6.34i)8-s + (−15.3 + 4.82i)10-s − 10.4i·11-s + 17.0·13-s + (5.18 + 16.4i)14-s + (5.47 − 15.0i)16-s + 25.7·17-s − 4.35i·19-s + (−26.3 + 18.4i)20-s + (−6.30 − 20.0i)22-s − 36.2i·23-s + ⋯ |
L(s) = 1 | + (0.953 − 0.300i)2-s + (0.819 − 0.573i)4-s − 1.60·5-s + 1.23i·7-s + (0.608 − 0.793i)8-s + (−1.53 + 0.482i)10-s − 0.953i·11-s + 1.31·13-s + (0.370 + 1.17i)14-s + (0.342 − 0.939i)16-s + 1.51·17-s − 0.229i·19-s + (−1.31 + 0.921i)20-s + (−0.286 − 0.909i)22-s − 1.57i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.709161814\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.709161814\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.90 + 0.601i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 5 | \( 1 + 8.03T + 25T^{2} \) |
| 7 | \( 1 - 8.62iT - 49T^{2} \) |
| 11 | \( 1 + 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 17.0T + 169T^{2} \) |
| 17 | \( 1 - 25.7T + 289T^{2} \) |
| 23 | \( 1 + 36.2iT - 529T^{2} \) |
| 29 | \( 1 + 25.4T + 841T^{2} \) |
| 31 | \( 1 + 14.6iT - 961T^{2} \) |
| 37 | \( 1 - 54.8T + 1.36e3T^{2} \) |
| 41 | \( 1 - 47.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 33.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 44.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 59.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 51.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 17.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 34.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 87.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 59.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 17.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 122. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 103.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 23.8T + 9.40e3T^{2} \) |
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
−10.61289740642367850444936676426, −9.182437723350999280703371867318, −8.278075848919455331692476450249, −7.58950211041548596626920100542, −6.22630036885078469875899451032, −5.66851746869652356925177648690, −4.40210921162550109647765530730, −3.54913690378174029829218192736, −2.73375208731842427964270081811, −0.856729131231814602381897589813,
1.25692541526800258337598889017, 3.41171515640735930784517684826, 3.80683101149178655730260196197, 4.65956484257238339050802314268, 5.87638085581800477848387535774, 7.09043276350191626612195755777, 7.64676093529098711878993531029, 8.096305012691841035564422407952, 9.659577616313373715190989791657, 10.82783696568753528131268949010