L(s) = 1 | + (0.667 + 1.15i)2-s + (3.10 − 5.38i)4-s − 9.00·5-s + (−14.2 + 11.8i)7-s + 18.9·8-s + (−6.01 − 10.4i)10-s − 29.3·11-s + (−21.1 − 36.6i)13-s + (−23.2 − 8.53i)14-s + (−12.1 − 21.1i)16-s + (2.56 + 4.45i)17-s + (−71.2 + 123. i)19-s + (−27.9 + 48.4i)20-s + (−19.6 − 33.9i)22-s − 178.·23-s + ⋯ |
L(s) = 1 | + (0.236 + 0.408i)2-s + (0.388 − 0.672i)4-s − 0.805·5-s + (−0.767 + 0.640i)7-s + 0.839·8-s + (−0.190 − 0.329i)10-s − 0.804·11-s + (−0.451 − 0.782i)13-s + (−0.443 − 0.162i)14-s + (−0.190 − 0.329i)16-s + (0.0366 + 0.0634i)17-s + (−0.860 + 1.49i)19-s + (−0.312 + 0.541i)20-s + (−0.189 − 0.329i)22-s − 1.61·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0107179 - 0.0761081i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0107179 - 0.0761081i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (14.2 - 11.8i)T \) |
good | 2 | \( 1 + (-0.667 - 1.15i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + 9.00T + 125T^{2} \) |
| 11 | \( 1 + 29.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + (21.1 + 36.6i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-2.56 - 4.45i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (71.2 - 123. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-109. + 189. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (73.9 - 128. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (21.2 - 36.8i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (83.7 + 145. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-121. + 210. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (38.2 + 66.2i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-181. - 314. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (60.7 - 105. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-321. - 556. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-81.4 + 141. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 833.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (62.4 + 108. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (421. + 729. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (566. - 982. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-248. + 429. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (128. - 223. i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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
−11.87114219156747943442862776495, −10.43835449731535308354626971287, −9.998884808457759584854170368584, −8.334301357186418864819666545412, −7.53535935682650092351204963705, −6.20818249264990422507439671205, −5.46771499247057968135582983349, −3.94660828790602406150478756485, −2.31218545420804853089894996977, −0.02847357616155964156196676473,
2.40700787148082735624464145810, 3.68257210179443301734478044839, 4.60412988968369922670501921626, 6.58805503881453651667673814081, 7.41593833020518472277904843814, 8.341131622671281210649403399653, 9.767061897478896142587042142490, 10.84708026185082502868971804974, 11.59354986411049906921736015047, 12.56789879356542255303250203456