L(s) = 1 | + (−4.5 + 2.59i)3-s + (−4 − 6.92i)4-s + (13.5 − 23.3i)9-s + (36 + 20.7i)12-s + 62.3i·13-s + (−31.9 + 55.4i)16-s + (135 + 77.9i)19-s + (62.5 + 108. i)25-s + 140. i·27-s + (−135 + 77.9i)31-s − 216·36-s + (55 − 95.2i)37-s + (−162 − 280. i)39-s + 520·43-s − 332. i·48-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.499i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)9-s + (0.866 + 0.499i)12-s + 1.33i·13-s + (−0.499 + 0.866i)16-s + (1.63 + 0.941i)19-s + (0.5 + 0.866i)25-s + 1.00i·27-s + (−0.782 + 0.451i)31-s − 36-s + (0.244 − 0.423i)37-s + (−0.665 − 1.15i)39-s + 1.84·43-s − 0.999i·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.775719 + 0.481258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.775719 + 0.481258i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.5 - 2.59i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 62.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-135 - 77.9i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 + (135 - 77.9i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-55 + 95.2i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 520T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (810 + 467. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-440 - 762. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 + (324 - 187. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (442 - 765. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.37e3iT - 9.12e5T^{2} \) |
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
−12.63873196349979223192622094904, −11.56650731253101578977590132992, −10.76051627745620709451713289877, −9.669719160580259528917779851105, −9.098921141686755449508402713263, −7.20485465144881848729034944209, −5.97264956388113791407930017405, −5.08864969674775856924329194285, −3.91029483423987447085577614417, −1.27087257537561382827861373003,
0.59114164568288093396210999289, 2.94844048321400508670640805001, 4.64221855032999634865341569835, 5.73414961016909173115037478452, 7.23527516182156214867234965693, 7.934377079934207405420513607946, 9.274441078007337872686550887648, 10.54790947751493021873182081754, 11.59891360461091762294151678547, 12.46510193891465516402282067257