| L(s) = 1 | + (16.9 − 12.3i)2-s + (8.34 + 25.6i)3-s + (96.2 − 296. i)4-s + (−380. − 276. i)5-s + (457. + 332. i)6-s + (39.7 − 122. i)7-s + (−1.18e3 − 3.66e3i)8-s + (−589. + 428. i)9-s − 9.86e3·10-s + (4.28e3 − 1.07e3i)11-s + 8.41e3·12-s + (7.49e3 − 5.44e3i)13-s + (−834. − 2.56e3i)14-s + (3.92e3 − 1.20e4i)15-s + (−3.30e4 − 2.40e4i)16-s + (6.11e3 + 4.43e3i)17-s + ⋯ |
| L(s) = 1 | + (1.49 − 1.08i)2-s + (0.178 + 0.549i)3-s + (0.752 − 2.31i)4-s + (−1.36 − 0.989i)5-s + (0.865 + 0.628i)6-s + (0.0438 − 0.134i)7-s + (−0.821 − 2.52i)8-s + (−0.269 + 0.195i)9-s − 3.11·10-s + (0.969 − 0.243i)11-s + 1.40·12-s + (0.946 − 0.687i)13-s + (−0.0812 − 0.250i)14-s + (0.300 − 0.923i)15-s + (−2.01 − 1.46i)16-s + (0.301 + 0.219i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.38039 - 3.04951i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.38039 - 3.04951i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-8.34 - 25.6i)T \) |
| 11 | \( 1 + (-4.28e3 + 1.07e3i)T \) |
| good | 2 | \( 1 + (-16.9 + 12.3i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (380. + 276. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-39.7 + 122. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (-7.49e3 + 5.44e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-6.11e3 - 4.43e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-1.30e4 - 4.00e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 5.83e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (1.61e4 - 4.95e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-2.52e5 + 1.83e5i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-4.16e4 + 1.28e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-4.27e4 - 1.31e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 5.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (2.57e5 + 7.91e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-4.44e5 + 3.23e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (1.20e5 - 3.70e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.10e6 - 1.53e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + 7.79e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.68e6 + 1.22e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (1.00e6 - 3.09e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (5.55e6 - 4.03e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (6.37e5 + 4.63e5i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 - 3.39e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-5.64e6 + 4.10e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−14.56386267985635663402923295477, −13.34508208614525778738384757525, −12.10852991876725446247935849937, −11.51547579592214863972600589604, −10.10781663933299706565331972035, −8.310860797547077897324215077404, −5.73890998649469880079815322145, −4.20358470597299667848068628254, −3.58675432108459408923405381873, −1.07662003446518065938428362466,
3.14001556591023668948486173147, 4.30693492554090026513372849942, 6.42646909334149922031595317273, 7.14930509357153491699004957787, 8.364797504748581973287189913492, 11.51438294416861679952273323083, 12.01092371469572062747475109333, 13.61345953138675998578115333362, 14.41329901831696070408551564538, 15.39407347896765208635230018796
