| L(s) = 1 | + (3.92 − 8.14i)2-s + (7.09 + 0.532i)3-s + (−11.0 − 13.8i)4-s + (43.9 − 142. i)5-s + (32.1 − 55.7i)6-s + (−72.9 + 42.1i)7-s + (407. − 93.0i)8-s + (−670. − 101. i)9-s + (−987. − 916. i)10-s + (1.14e3 − 1.42e3i)11-s + (−71.2 − 104. i)12-s + (798. − 740. i)13-s + (56.9 + 759. i)14-s + (387. − 987. i)15-s + (1.09e3 − 4.79e3i)16-s + (−9.19e3 + 2.83e3i)17-s + ⋯ |
| L(s) = 1 | + (0.490 − 1.01i)2-s + (0.262 + 0.0197i)3-s + (−0.173 − 0.217i)4-s + (0.351 − 1.13i)5-s + (0.149 − 0.258i)6-s + (−0.212 + 0.122i)7-s + (0.796 − 0.181i)8-s + (−0.920 − 0.138i)9-s + (−0.987 − 0.916i)10-s + (0.856 − 1.07i)11-s + (−0.0412 − 0.0604i)12-s + (0.363 − 0.337i)13-s + (0.0207 + 0.276i)14-s + (0.114 − 0.292i)15-s + (0.267 − 1.17i)16-s + (−1.87 + 0.577i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.17058 - 2.24664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17058 - 2.24664i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.48e4 + 2.67e4i)T \) |
| good | 2 | \( 1 + (-3.92 + 8.14i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-7.09 - 0.532i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (-43.9 + 142. i)T + (-1.29e4 - 8.80e3i)T^{2} \) |
| 7 | \( 1 + (72.9 - 42.1i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.14e3 + 1.42e3i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (-798. + 740. i)T + (3.60e5 - 4.81e6i)T^{2} \) |
| 17 | \( 1 + (9.19e3 - 2.83e3i)T + (1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (423. + 2.81e3i)T + (-4.49e7 + 1.38e7i)T^{2} \) |
| 23 | \( 1 + (-7.07e3 - 1.80e4i)T + (-1.08e8 + 1.00e8i)T^{2} \) |
| 29 | \( 1 + (-1.73e4 + 1.30e3i)T + (5.88e8 - 8.86e7i)T^{2} \) |
| 31 | \( 1 + (-4.49e4 + 3.06e4i)T + (3.24e8 - 8.26e8i)T^{2} \) |
| 37 | \( 1 + (-8.45e4 - 4.87e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-2.64e3 - 1.27e3i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-5.36e4 - 6.72e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-1.25e5 - 1.16e5i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (4.58e4 - 2.01e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-2.41e4 + 3.54e4i)T + (-1.88e10 - 4.79e10i)T^{2} \) |
| 67 | \( 1 + (-1.19e5 + 1.80e4i)T + (8.64e10 - 2.66e10i)T^{2} \) |
| 71 | \( 1 + (4.96e5 + 1.94e5i)T + (9.39e10 + 8.71e10i)T^{2} \) |
| 73 | \( 1 + (3.70e5 + 3.99e5i)T + (-1.13e10 + 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-2.49e4 - 4.31e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (6.06e4 - 8.08e5i)T + (-3.23e11 - 4.87e10i)T^{2} \) |
| 89 | \( 1 + (2.56e5 + 1.91e4i)T + (4.91e11 + 7.40e10i)T^{2} \) |
| 97 | \( 1 + (1.37e5 - 1.72e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.54110818473312551786344826565, −13.32819069647878875869301469734, −11.83612843876109686839929611742, −11.08823917812096542920332776503, −9.312296967395234051475310681791, −8.398556553060604183820765245219, −6.08616901233166624392216079246, −4.38815099214087826092630180560, −2.86557991416073943702624113481, −1.08101320324286086479859431454,
2.42039312674184992126291178368, 4.52505173380167509546563609049, 6.43369171720076115134687435504, 6.87010943945563629618405453764, 8.646609346395234931336019464062, 10.29157476149715988272174422462, 11.44850560046813991256314545375, 13.31187916587623431248082769782, 14.38246482068957037279987110352, 14.75524265157364872841079272637
