| L(s) = 1 | + (0.103 − 0.385i)2-s + (3.32 + 1.92i)4-s + (2.38 + 8.89i)5-s + (6.18 − 10.7i)7-s + (2.21 − 2.21i)8-s + 3.67·10-s − 18.0i·11-s + (0.292 + 1.09i)13-s + (−3.49 − 3.49i)14-s + (7.05 + 12.2i)16-s + (16.1 + 4.32i)17-s + (3.09 + 11.5i)19-s + (−9.15 + 34.1i)20-s + (−6.96 − 1.86i)22-s + (−9.42 + 9.42i)23-s + ⋯ |
| L(s) = 1 | + (0.0516 − 0.192i)2-s + (0.831 + 0.480i)4-s + (0.476 + 1.77i)5-s + (0.883 − 1.52i)7-s + (0.276 − 0.276i)8-s + 0.367·10-s − 1.64i·11-s + (0.0225 + 0.0840i)13-s + (−0.249 − 0.249i)14-s + (0.440 + 0.763i)16-s + (0.949 + 0.254i)17-s + (0.162 + 0.608i)19-s + (−0.457 + 1.70i)20-s + (−0.316 − 0.0848i)22-s + (−0.409 + 0.409i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.321i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.44733 + 0.403525i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.44733 + 0.403525i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 37 | \( 1 + (36.0 + 8.39i)T \) |
| good | 2 | \( 1 + (-0.103 + 0.385i)T + (-3.46 - 2i)T^{2} \) |
| 5 | \( 1 + (-2.38 - 8.89i)T + (-21.6 + 12.5i)T^{2} \) |
| 7 | \( 1 + (-6.18 + 10.7i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + 18.0iT - 121T^{2} \) |
| 13 | \( 1 + (-0.292 - 1.09i)T + (-146. + 84.5i)T^{2} \) |
| 17 | \( 1 + (-16.1 - 4.32i)T + (250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-3.09 - 11.5i)T + (-312. + 180.5i)T^{2} \) |
| 23 | \( 1 + (9.42 - 9.42i)T - 529iT^{2} \) |
| 29 | \( 1 + (-1.29 - 1.29i)T + 841iT^{2} \) |
| 31 | \( 1 + (-12.9 - 12.9i)T + 961iT^{2} \) |
| 41 | \( 1 + (38.7 + 22.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (9.25 - 9.25i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 9.05T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-11.8 - 20.4i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (23.8 + 6.38i)T + (3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (37.7 - 10.1i)T + (3.22e3 - 1.86e3i)T^{2} \) |
| 67 | \( 1 + (12.2 + 7.07i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-18.3 + 31.8i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 - 88.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (19.9 + 74.4i)T + (-5.40e3 + 3.12e3i)T^{2} \) |
| 83 | \( 1 + (-27.0 - 46.8i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (7.72 - 28.8i)T + (-6.85e3 - 3.96e3i)T^{2} \) |
| 97 | \( 1 + (-120. + 120. i)T - 9.40e3iT^{2} \) |
| 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
−11.18021606677197395950238101830, −10.59139438242699968757922396785, −10.15362823595709903012030290086, −8.195535929047766783981660475779, −7.48654436970403241984359610416, −6.70971016394270851850859975908, −5.77155287897884587879266213464, −3.71856177483881110593926467561, −3.16426357051068554708025127946, −1.57808472206802188294343877362,
1.47713292723367671383922634107, 2.27408260784380265289012075760, 4.87952707813036797124542988383, 5.12911628157951561872527088100, 6.18171893971583011167542828637, 7.62955458576215576680573163150, 8.510118929259585946199567751290, 9.413597236115820955028896179710, 10.15457252241849532284374521014, 11.76978300797426083383368167785
