| L(s) = 1 | + (1.99 − 0.121i)2-s + (1.09 − 1.33i)3-s + (3.97 − 0.486i)4-s + (0.0915 + 0.301i)5-s + (2.03 − 2.80i)6-s + (−2.11 + 10.6i)7-s + (7.86 − 1.45i)8-s + (−0.585 − 2.94i)9-s + (0.219 + 0.591i)10-s + (8.45 + 0.832i)11-s + (3.71 − 5.85i)12-s + (11.2 + 3.41i)13-s + (−2.92 + 21.4i)14-s + (0.504 + 0.209i)15-s + (15.5 − 3.86i)16-s + (0.851 + 2.05i)17-s + ⋯ |
| L(s) = 1 | + (0.998 − 0.0609i)2-s + (0.366 − 0.446i)3-s + (0.992 − 0.121i)4-s + (0.0183 + 0.0603i)5-s + (0.338 − 0.467i)6-s + (−0.302 + 1.51i)7-s + (0.983 − 0.181i)8-s + (−0.0650 − 0.326i)9-s + (0.0219 + 0.0591i)10-s + (0.768 + 0.0757i)11-s + (0.309 − 0.487i)12-s + (0.865 + 0.262i)13-s + (−0.209 + 1.53i)14-s + (0.0336 + 0.0139i)15-s + (0.970 − 0.241i)16-s + (0.0500 + 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.73248 - 0.0282539i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.73248 - 0.0282539i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 + 0.121i)T \) |
| 3 | \( 1 + (-1.09 + 1.33i)T \) |
| good | 5 | \( 1 + (-0.0915 - 0.301i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (2.11 - 10.6i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-8.45 - 0.832i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-11.2 - 3.41i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-0.851 - 2.05i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-12.8 + 6.86i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (33.4 + 22.3i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (3.21 - 0.316i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (21.8 - 21.8i)T - 961iT^{2} \) |
| 37 | \( 1 + (59.1 + 31.6i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (31.1 + 20.7i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-23.6 - 28.8i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (6.57 + 15.8i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-16.3 - 1.60i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-10.9 - 36.2i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (68.8 - 83.9i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (13.6 + 11.1i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-54.6 - 10.8i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-95.9 + 19.0i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (49.7 - 120. i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (20.3 + 38.0i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (36.5 + 54.6i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (77.6 + 77.6i)T + 9.40e3iT^{2} \) |
| 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
−11.52125402529200302369914088800, −10.39072175787727262967988284363, −9.098836820868767698223810247814, −8.413838858066441390562764928712, −7.01122318256701921243394848143, −6.23805608423886737953908241145, −5.41532716645983793985215897623, −3.96866754027274999848302288266, −2.85126979822719674919871277627, −1.75188920417421724889404255761,
1.46947688432465463740461733597, 3.53123218074915881944761381225, 3.77697327453776933582901280490, 5.09461734794575307444397528506, 6.28131956828920864348960738611, 7.23867540238063039486691487465, 8.095481056042399590976542256955, 9.507770342935835299089744914371, 10.39229747778377216682862017172, 11.12754465018643107738154856507
