| L(s) = 1 | + (3.00 + 6.23i)2-s + (14.9 + 21.9i)3-s + (10.0 − 12.5i)4-s + (96.6 − 104. i)5-s + (−91.8 + 159. i)6-s + (401. − 231. i)7-s + (540. + 123. i)8-s + (9.10 − 23.1i)9-s + (939. + 289. i)10-s + (−377. − 473. i)11-s + (425. + 31.8i)12-s + (−3.63e3 + 1.12e3i)13-s + (2.65e3 + 1.80e3i)14-s + (3.72e3 + 561. i)15-s + (624. + 2.73e3i)16-s + (−3.85e3 + 3.57e3i)17-s + ⋯ |
| L(s) = 1 | + (0.375 + 0.779i)2-s + (0.553 + 0.811i)3-s + (0.156 − 0.196i)4-s + (0.773 − 0.833i)5-s + (−0.425 + 0.736i)6-s + (1.17 − 0.675i)7-s + (1.05 + 0.240i)8-s + (0.0124 − 0.0318i)9-s + (0.939 + 0.289i)10-s + (−0.283 − 0.355i)11-s + (0.246 + 0.0184i)12-s + (−1.65 + 0.510i)13-s + (0.966 + 0.658i)14-s + (1.10 + 0.166i)15-s + (0.152 + 0.668i)16-s + (−0.784 + 0.728i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.96782 + 1.21846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.96782 + 1.21846i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.69e4 - 6.41e4i)T \) |
| good | 2 | \( 1 + (-3.00 - 6.23i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-14.9 - 21.9i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (-96.6 + 104. i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-401. + 231. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (377. + 473. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (3.63e3 - 1.12e3i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (3.85e3 - 3.57e3i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (648. - 254. i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (-321. + 48.5i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (2.03e4 - 2.98e4i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (1.49e3 - 1.99e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (4.61e4 + 2.66e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-5.72e4 + 2.75e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (1.84e4 - 2.31e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (9.38e4 + 2.89e4i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (1.68e4 + 7.39e4i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-3.81e5 + 2.85e4i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (-1.64e5 - 4.19e5i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (-5.22e4 + 3.46e5i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (-1.61e5 - 5.25e5i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (3.57e5 + 6.19e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (1.64e5 - 1.11e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (6.10e5 + 8.95e5i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (7.83e5 + 9.82e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.57450747481109882860332186922, −14.30888005270982230342223721848, −12.87509121553425869759261917953, −10.98203734959818202172876018242, −9.854099635587019265701419047999, −8.584726666507072736176195628789, −7.10693772476155578702960521031, −5.27137803342167272675102602722, −4.44503946418297535595589173710, −1.72132754023434027453729135947,
2.15329572256325633154547788450, 2.44775041830303915312100556359, 4.95783496280343830760082886075, 7.07500684779297905008255364331, 7.992548094013260334515376751127, 9.918262319161856125867060300487, 11.17284060171083475588997467954, 12.24610487508193318604187287894, 13.31822896982544652819155513409, 14.23814644563456645397688572188
