| L(s) = 1 | + (9.06 + 39.7i)2-s + (57.6 + 53.4i)3-s + (−1.03e3 + 498. i)4-s + (−704. + 106. i)5-s + (−1.60e3 + 2.77e3i)6-s + (−4.19e3 − 7.25e3i)7-s + (−1.61e4 − 2.02e4i)8-s + (−1.00e3 − 1.34e4i)9-s + (−1.06e4 − 2.70e4i)10-s + (6.76e4 + 3.25e4i)11-s + (−8.63e4 − 2.66e4i)12-s + (−3.09e4 + 7.89e4i)13-s + (2.50e5 − 2.32e5i)14-s + (−4.62e4 − 3.15e4i)15-s + (2.92e5 − 3.66e5i)16-s + (−4.69e5 − 7.07e4i)17-s + ⋯ |
| L(s) = 1 | + (0.400 + 1.75i)2-s + (0.410 + 0.381i)3-s + (−2.02 + 0.973i)4-s + (−0.503 + 0.0759i)5-s + (−0.504 + 0.874i)6-s + (−0.659 − 1.14i)7-s + (−1.39 − 1.75i)8-s + (−0.0512 − 0.683i)9-s + (−0.335 − 0.854i)10-s + (1.39 + 0.670i)11-s + (−1.20 − 0.370i)12-s + (−0.300 + 0.766i)13-s + (1.74 − 1.61i)14-s + (−0.235 − 0.160i)15-s + (1.11 − 1.39i)16-s + (−1.36 − 0.205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.442327 - 0.0774512i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.442327 - 0.0774512i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-5.06e6 + 2.18e7i)T \) |
| good | 2 | \( 1 + (-9.06 - 39.7i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-57.6 - 53.4i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (704. - 106. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (4.19e3 + 7.25e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-6.76e4 - 3.25e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (3.09e4 - 7.89e4i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (4.69e5 + 7.07e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-7.65e4 + 1.02e6i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (6.86e5 - 4.67e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-6.88e5 + 6.38e5i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (-1.11e6 - 3.44e5i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (1.02e7 - 1.76e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (4.16e6 + 1.82e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (-8.76e6 + 4.22e6i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (1.74e7 + 4.44e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (-6.52e7 + 8.17e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (7.86e7 - 2.42e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (4.57e6 - 6.10e7i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (1.66e8 + 1.13e8i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (1.28e8 - 3.27e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (-1.12e8 - 1.94e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (9.58e7 + 8.88e7i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (3.82e8 + 3.54e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (6.42e8 + 3.09e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−14.13698074185576634003750338151, −13.42453938093574061798860062344, −11.80227519519930131193585230352, −9.609195172870322566406035639983, −8.785068978967175848925582089188, −6.92611742994790747413856568225, −6.79717379549018116063431601953, −4.46767329437834930809239318619, −3.80927875242250445103606506437, −0.12994692876795777296641149521,
1.66217521558365577482608636894, 2.85180738812591901207992640808, 4.07558491904612470090959400062, 5.88315280345321557602437370287, 8.317998747835684087793247556007, 9.339157418327795514707024019956, 10.70385142584539076237516776835, 11.92380585739791542426544998985, 12.52534492229624373129776600509, 13.65646422066371916655590328949
