L(s) = 1 | + (0.222 + 0.385i)2-s + (−0.900 + 1.56i)3-s + (0.400 − 0.694i)4-s − 0.801·6-s + 0.801·8-s + (−1.12 − 1.94i)9-s + (0.722 + 1.25i)12-s + 0.445·13-s + (−0.222 − 0.385i)16-s + (0.623 − 1.07i)17-s + (0.499 − 0.866i)18-s + (0.623 + 1.07i)19-s + (0.900 + 1.56i)23-s + (−0.722 + 1.25i)24-s + (−0.5 + 0.866i)25-s + (0.0990 + 0.171i)26-s + ⋯ |
L(s) = 1 | + (0.222 + 0.385i)2-s + (−0.900 + 1.56i)3-s + (0.400 − 0.694i)4-s − 0.801·6-s + 0.801·8-s + (−1.12 − 1.94i)9-s + (0.722 + 1.25i)12-s + 0.445·13-s + (−0.222 − 0.385i)16-s + (0.623 − 1.07i)17-s + (0.499 − 0.866i)18-s + (0.623 + 1.07i)19-s + (0.900 + 1.56i)23-s + (−0.722 + 1.25i)24-s + (−0.5 + 0.866i)25-s + (0.0990 + 0.171i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.137403808\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.137403808\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - 0.445T + T^{2} \) |
| 17 | \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 - 1.24T + T^{2} \) |
| 47 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - 1.80T + T^{2} \) |
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
−9.601281576853198179698593941138, −9.224428101381131020868302177269, −7.82040818671948578844971855507, −6.99548593462048006892645907170, −5.99624923973918533493469440353, −5.41406783742914857800685793023, −5.04401577427435506798130622400, −3.93556652893087677963129953892, −3.15988414154023773290477849106, −1.28325315793794628384740804361,
1.03645408969807067672409411965, 2.13557826763703809556282800369, 2.96112448191744686442098279712, 4.24639969694681809920604621951, 5.26922037028828391186609481251, 6.26609944412538972041720521459, 6.71956607652015870626490659319, 7.52770341845093308909208361251, 8.109312287390145997948611799640, 8.852285579607738733376073028902