L(s) = 1 | + (−0.5 + 0.866i)3-s + (1.37 + 2.37i)5-s + (2.64 + 0.0585i)7-s + (−0.499 − 0.866i)9-s + (−0.771 + 1.33i)11-s − 6.03·13-s − 2.74·15-s + (−3.74 + 6.48i)17-s + (3.01 + 5.22i)19-s + (−1.37 + 2.26i)21-s + (−3.74 − 6.48i)23-s + (−1.27 + 2.20i)25-s + 0.999·27-s − 1.25·29-s + (2.64 − 4.58i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.614 + 1.06i)5-s + (0.999 + 0.0221i)7-s + (−0.166 − 0.288i)9-s + (−0.232 + 0.403i)11-s − 1.67·13-s − 0.709·15-s + (−0.908 + 1.57i)17-s + (0.692 + 1.19i)19-s + (−0.299 + 0.493i)21-s + (−0.781 − 1.35i)23-s + (−0.254 + 0.440i)25-s + 0.192·27-s − 0.232·29-s + (0.475 − 0.822i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.297319360\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.297319360\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.64 - 0.0585i)T \) |
good | 5 | \( 1 + (-1.37 - 2.37i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.771 - 1.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + (3.74 - 6.48i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.01 - 5.22i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.74 + 6.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 + (-2.64 + 4.58i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.47 - 4.28i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 43 | \( 1 - 3.45T + 43T^{2} \) |
| 47 | \( 1 + (-4.74 - 8.22i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.91 - 3.32i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.77 - 4.80i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.29 + 12.6i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.01 + 3.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + (6.27 - 10.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.89 + 6.75i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6.52T + 83T^{2} \) |
| 89 | \( 1 + (-4.74 - 8.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 1.54T + 97T^{2} \) |
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
−10.29178208232218161816083678905, −9.359034410989247597227491805188, −8.179180810419348444424026410392, −7.58359192853571980436258670548, −6.49325965605160561106345423238, −5.87543163125202022953324035001, −4.80014012639828417725769777084, −4.12253074314012138804653735825, −2.67268520137177101874091352703, −1.90616643814755816392225714455,
0.52247170458367732077386582886, 1.80029810268692296238810527881, 2.74080257619574099809617749331, 4.59826431943075128251551866583, 5.08708986527030683706868069964, 5.64985879175433730472008626767, 7.13673773037082681928275762718, 7.45024104858611370257739218271, 8.572848937413789245959594309302, 9.239696450372316205959153080451