| L(s) = 1 | + (−3.74 + 7.78i)2-s + (23.1 + 48.0i)3-s + (113. + 141. i)4-s + (−1.13e3 + 259. i)5-s − 460.·6-s − 3.46e3i·7-s + (−3.68e3 + 840. i)8-s + (2.31e3 − 2.90e3i)9-s + (2.23e3 − 9.80e3i)10-s + (3.83e3 − 4.80e3i)11-s + (−4.19e3 + 8.71e3i)12-s + (−115. − 507. i)13-s + (2.69e4 + 1.29e4i)14-s + (−3.87e4 − 4.85e4i)15-s + (−3.07e3 + 1.34e4i)16-s + (−1.58e4 + 6.96e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.234 + 0.486i)2-s + (0.285 + 0.593i)3-s + (0.441 + 0.554i)4-s + (−1.81 + 0.414i)5-s − 0.355·6-s − 1.44i·7-s + (−0.899 + 0.205i)8-s + (0.353 − 0.443i)9-s + (0.223 − 0.980i)10-s + (0.261 − 0.328i)11-s + (−0.202 + 0.420i)12-s + (−0.00405 − 0.0177i)13-s + (0.702 + 0.338i)14-s + (−0.764 − 0.958i)15-s + (−0.0469 + 0.205i)16-s + (−0.190 + 0.833i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.630 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.622711 - 0.296564i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.622711 - 0.296564i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.24e5 + 3.37e6i)T \) |
| good | 2 | \( 1 + (3.74 - 7.78i)T + (-159. - 200. i)T^{2} \) |
| 3 | \( 1 + (-23.1 - 48.0i)T + (-4.09e3 + 5.12e3i)T^{2} \) |
| 5 | \( 1 + (1.13e3 - 259. i)T + (3.51e5 - 1.69e5i)T^{2} \) |
| 7 | \( 1 + 3.46e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (-3.83e3 + 4.80e3i)T + (-4.76e7 - 2.08e8i)T^{2} \) |
| 13 | \( 1 + (115. + 507. i)T + (-7.34e8 + 3.53e8i)T^{2} \) |
| 17 | \( 1 + (1.58e4 - 6.96e4i)T + (-6.28e9 - 3.02e9i)T^{2} \) |
| 19 | \( 1 + (-4.17e4 + 3.33e4i)T + (3.77e9 - 1.65e10i)T^{2} \) |
| 23 | \( 1 + (-5.70e4 + 7.15e4i)T + (-1.74e10 - 7.63e10i)T^{2} \) |
| 29 | \( 1 + (-1.11e4 + 2.31e4i)T + (-3.11e11 - 3.91e11i)T^{2} \) |
| 31 | \( 1 + (1.53e6 + 7.40e5i)T + (5.31e11 + 6.66e11i)T^{2} \) |
| 37 | \( 1 + 2.51e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (1.47e6 + 7.10e5i)T + (4.97e12 + 6.24e12i)T^{2} \) |
| 47 | \( 1 + (3.16e6 + 3.96e6i)T + (-5.29e12 + 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-1.11e6 + 4.88e6i)T + (-5.60e13 - 2.70e13i)T^{2} \) |
| 59 | \( 1 + (4.67e6 - 2.04e7i)T + (-1.32e14 - 6.37e13i)T^{2} \) |
| 61 | \( 1 + (6.02e6 + 1.25e7i)T + (-1.19e14 + 1.49e14i)T^{2} \) |
| 67 | \( 1 + (-1.38e7 - 1.73e7i)T + (-9.03e13 + 3.95e14i)T^{2} \) |
| 71 | \( 1 + (-2.99e7 + 2.38e7i)T + (1.43e14 - 6.29e14i)T^{2} \) |
| 73 | \( 1 + (7.01e6 - 1.60e6i)T + (7.26e14 - 3.49e14i)T^{2} \) |
| 79 | \( 1 + 4.74e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (3.10e7 - 1.49e7i)T + (1.40e15 - 1.76e15i)T^{2} \) |
| 89 | \( 1 + (2.33e7 + 4.85e7i)T + (-2.45e15 + 3.07e15i)T^{2} \) |
| 97 | \( 1 + (-1.09e8 + 1.37e8i)T + (-1.74e15 - 7.64e15i)T^{2} \) |
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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
−14.56892784345685006995085114066, −12.72241843357380535610400516691, −11.47853447393006493413166912329, −10.58792423847000689832067994515, −8.741650027243694281799206201568, −7.55500961197697428714151383949, −6.88602923451375069132634416315, −3.99317859380041626908866337801, −3.54751477057883523512459600318, −0.28370854153701518538833498806,
1.44076165173520381432732830796, 2.99145958669976239639215885427, 5.01866242370516979918003880846, 6.94667123920799555256512412339, 8.163183954568981040979173240439, 9.332990442708727183571135683140, 11.16357387991859766880236926503, 11.95677632833709364413590588888, 12.67544794029265344662430985190, 14.64333611432635840681013363963
