L(s) = 1 | + (1.22 + 0.707i)2-s + (−0.866 − 1.5i)3-s + (0.999 + 1.73i)4-s − 2.44i·6-s + (6.98 + 0.440i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−3.76 − 6.52i)11-s + (1.73 − 2.99i)12-s − 21.3·13-s + (8.24 + 5.47i)14-s + (−2.00 + 3.46i)16-s + (−10.4 − 18.1i)17-s + (−3.67 + 2.12i)18-s + (−20.8 − 12.0i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.288 − 0.5i)3-s + (0.249 + 0.433i)4-s − 0.408i·6-s + (0.998 + 0.0628i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.342 − 0.593i)11-s + (0.144 − 0.249i)12-s − 1.64·13-s + (0.588 + 0.391i)14-s + (−0.125 + 0.216i)16-s + (−0.617 − 1.06i)17-s + (−0.204 + 0.117i)18-s + (−1.09 − 0.634i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.037716347\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.037716347\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.98 - 0.440i)T \) |
good | 11 | \( 1 + (3.76 + 6.52i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 21.3T + 169T^{2} \) |
| 17 | \( 1 + (10.4 + 18.1i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (20.8 + 12.0i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-4.83 - 2.79i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 9.96T + 841T^{2} \) |
| 31 | \( 1 + (-5.70 + 3.29i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (19.3 + 11.1i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 51.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 34.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-38.8 + 67.2i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (43.1 - 24.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (72.9 - 42.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (72.4 + 41.8i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-57.6 + 33.2i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 68.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + (43.8 + 75.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-49.3 + 85.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 26.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + (12.0 + 6.95i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 3.69T + 9.40e3T^{2} \) |
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
−9.241298797824034928101588524161, −8.398012087978938341500195963377, −7.52678625110899114020583536147, −6.98222334858732540068110178940, −5.95611071256784825090156714005, −4.93193661111428446931520644038, −4.59537048982730278141471112384, −2.89152822241151148303661801440, −2.04318776990961335925221467856, −0.24348534115068906251376761850,
1.70663467123549663616412153087, 2.65454476199106007316879378269, 4.17679448787918721442164284331, 4.62444393767140573916827596493, 5.45310540579332319269622801971, 6.47458527134352777529571304464, 7.47804731499124127510910826164, 8.352201940198699637869669455129, 9.386590286757908073350445400760, 10.35624548403274461840548590624