L(s) = 1 | + (0.707 − 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (−5.12 − 2.95i)5-s − 2.44i·6-s − 2.82·8-s + (1.5 − 2.59i)9-s + (−7.24 + 4.18i)10-s + (0.878 + 1.52i)11-s + (−2.99 − 1.73i)12-s − 18.7i·13-s − 10.2·15-s + (−2.00 + 3.46i)16-s + (−20.3 + 11.7i)17-s + (−2.12 − 3.67i)18-s + (−19.9 − 11.5i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.5 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−1.02 − 0.591i)5-s − 0.408i·6-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.724 + 0.418i)10-s + (0.0798 + 0.138i)11-s + (−0.249 − 0.144i)12-s − 1.44i·13-s − 0.682·15-s + (−0.125 + 0.216i)16-s + (−1.19 + 0.690i)17-s + (−0.117 − 0.204i)18-s + (−1.05 − 0.606i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0398569 - 1.25738i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0398569 - 1.25738i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (5.12 + 2.95i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-0.878 - 1.52i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 18.7iT - 169T^{2} \) |
| 17 | \( 1 + (20.3 - 11.7i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (19.9 + 11.5i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (9.36 - 16.2i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 30T + 841T^{2} \) |
| 31 | \( 1 + (7.45 - 4.30i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-35.4 + 61.4i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 41.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-33.5 - 19.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-18.5 - 32.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-84.4 + 48.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-14.4 - 8.36i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (30.4 + 52.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 110.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-49.1 + 28.3i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-34.9 + 60.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 6.43iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (36.4 + 21.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 51.7iT - 9.40e3T^{2} \) |
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
−11.15144173605365713722016822109, −10.39134383748512289814300719610, −9.030068745514118689854526263715, −8.350372312551556340697704115193, −7.38984346726022459837412422355, −5.97725140538962698289242091342, −4.56979915207340690639701127954, −3.72876978288576317481396792878, −2.32740498244718668295045756806, −0.50554266829983590438363000804,
2.56449419282933204546869393149, 3.99420022922746849639968796605, 4.55537043851791870286105648100, 6.39320548256240311305814639110, 7.05234682051587544273261600142, 8.193330988592548779873497908426, 8.862402404848567803043637181799, 10.08447283911005827431450693736, 11.29706377751817352501183803269, 11.87697628492047624680901191863