| L(s) = 1 | + (2.55 + 11.2i)2-s + (−51.6 − 47.9i)3-s + (−3.60 + 1.73i)4-s + (81.6 − 12.3i)5-s + (405. − 701. i)6-s + (118. + 205. i)7-s + (888. + 1.11e3i)8-s + (208. + 2.77e3i)9-s + (346. + 882. i)10-s + (−7.76e3 − 3.74e3i)11-s + (269. + 83.0i)12-s + (−712. + 1.81e3i)13-s + (−1.99e3 + 1.85e3i)14-s + (−4.81e3 − 3.27e3i)15-s + (−1.05e4 + 1.31e4i)16-s + (−2.05e4 − 3.09e3i)17-s + ⋯ |
| L(s) = 1 | + (0.225 + 0.990i)2-s + (−1.10 − 1.02i)3-s + (−0.0281 + 0.0135i)4-s + (0.292 − 0.0440i)5-s + (0.765 − 1.32i)6-s + (0.130 + 0.226i)7-s + (0.613 + 0.769i)8-s + (0.0951 + 1.26i)9-s + (0.109 + 0.279i)10-s + (−1.75 − 0.847i)11-s + (0.0450 + 0.0138i)12-s + (−0.0899 + 0.229i)13-s + (−0.194 + 0.180i)14-s + (−0.367 − 0.250i)15-s + (−0.642 + 0.805i)16-s + (−1.01 − 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0124644 - 0.189372i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0124644 - 0.189372i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.29e5 + 5.05e5i)T \) |
| good | 2 | \( 1 + (-2.55 - 11.2i)T + (-115. + 55.5i)T^{2} \) |
| 3 | \( 1 + (51.6 + 47.9i)T + (163. + 2.18e3i)T^{2} \) |
| 5 | \( 1 + (-81.6 + 12.3i)T + (7.46e4 - 2.30e4i)T^{2} \) |
| 7 | \( 1 + (-118. - 205. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (7.76e3 + 3.74e3i)T + (1.21e7 + 1.52e7i)T^{2} \) |
| 13 | \( 1 + (712. - 1.81e3i)T + (-4.59e7 - 4.26e7i)T^{2} \) |
| 17 | \( 1 + (2.05e4 + 3.09e3i)T + (3.92e8 + 1.20e8i)T^{2} \) |
| 19 | \( 1 + (2.90e3 - 3.88e4i)T + (-8.83e8 - 1.33e8i)T^{2} \) |
| 23 | \( 1 + (-2.16e4 + 1.47e4i)T + (1.24e9 - 3.16e9i)T^{2} \) |
| 29 | \( 1 + (1.86e4 - 1.73e4i)T + (1.28e9 - 1.72e10i)T^{2} \) |
| 31 | \( 1 + (2.19e5 + 6.75e4i)T + (2.27e10 + 1.54e10i)T^{2} \) |
| 37 | \( 1 + (1.80e5 - 3.12e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-1.05e5 - 4.63e5i)T + (-1.75e11 + 8.45e10i)T^{2} \) |
| 47 | \( 1 + (-5.55e5 + 2.67e5i)T + (3.15e11 - 3.96e11i)T^{2} \) |
| 53 | \( 1 + (-1.46e4 - 3.74e4i)T + (-8.61e11 + 7.99e11i)T^{2} \) |
| 59 | \( 1 + (-5.02e5 + 6.30e5i)T + (-5.53e11 - 2.42e12i)T^{2} \) |
| 61 | \( 1 + (5.27e5 - 1.62e5i)T + (2.59e12 - 1.77e12i)T^{2} \) |
| 67 | \( 1 + (-1.31e5 + 1.75e6i)T + (-5.99e12 - 9.03e11i)T^{2} \) |
| 71 | \( 1 + (-2.82e6 - 1.92e6i)T + (3.32e12 + 8.46e12i)T^{2} \) |
| 73 | \( 1 + (4.23e5 - 1.07e6i)T + (-8.09e12 - 7.51e12i)T^{2} \) |
| 79 | \( 1 + (2.70e6 + 4.67e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (2.84e6 + 2.63e6i)T + (2.02e12 + 2.70e13i)T^{2} \) |
| 89 | \( 1 + (1.63e6 + 1.51e6i)T + (3.30e12 + 4.41e13i)T^{2} \) |
| 97 | \( 1 + (1.20e7 + 5.78e6i)T + (5.03e13 + 6.31e13i)T^{2} \) |
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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
−15.25467790140690810003661801055, −13.74906274700024128292855247823, −12.94293376747997538213640438501, −11.52900995523186989066125272310, −10.59069689970244598933324505201, −8.237131821041156459669739836833, −7.15697224909359144117350144548, −5.94624859558784682256428830270, −5.28403283923017076040476236506, −1.97010641260621611148126965558,
0.07681339796149662841938692707, 2.38657620355862235782850955254, 4.26677668645603940679087218577, 5.35749079183974336398066901166, 7.23660455639035308125105534789, 9.575349503166492227858177429435, 10.70322485384126558390417746605, 10.99651585010813015859077031287, 12.44613329550843836553906877966, 13.37640120479795427562460163017
