| L(s) = 1 | + (0.974 + 1.34i)2-s + (−1.09 + 2.79i)3-s + (0.386 − 1.18i)4-s + (−0.410 + 0.565i)5-s + (−4.81 + 1.24i)6-s + (0.806 − 2.48i)7-s + (8.28 − 2.69i)8-s + (−6.59 − 6.12i)9-s − 1.15·10-s + (4.26 − 10.1i)11-s + (2.89 + 2.38i)12-s + (−13.8 + 10.0i)13-s + (4.11 − 1.33i)14-s + (−1.12 − 1.76i)15-s + (7.63 + 5.54i)16-s + (−9.47 + 13.0i)17-s + ⋯ |
| L(s) = 1 | + (0.487 + 0.670i)2-s + (−0.365 + 0.930i)3-s + (0.0966 − 0.297i)4-s + (−0.0821 + 0.113i)5-s + (−0.802 + 0.208i)6-s + (0.115 − 0.354i)7-s + (1.03 − 0.336i)8-s + (−0.732 − 0.680i)9-s − 0.115·10-s + (0.387 − 0.921i)11-s + (0.241 + 0.198i)12-s + (−1.06 + 0.773i)13-s + (0.293 − 0.0954i)14-s + (−0.0751 − 0.117i)15-s + (0.477 + 0.346i)16-s + (−0.557 + 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.999672 + 0.605265i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.999672 + 0.605265i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.09 - 2.79i)T \) |
| 11 | \( 1 + (-4.26 + 10.1i)T \) |
| good | 2 | \( 1 + (-0.974 - 1.34i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (0.410 - 0.565i)T + (-7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (-0.806 + 2.48i)T + (-39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (13.8 - 10.0i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (9.47 - 13.0i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (4.92 + 15.1i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 - 23.1iT - 529T^{2} \) |
| 29 | \( 1 + (5.10 + 1.65i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-3.28 + 2.38i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-19.6 + 60.4i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (64.1 - 20.8i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + 22.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-70.2 + 22.8i)T + (1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-25.1 - 34.6i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (-27.3 - 8.88i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-37.4 - 27.2i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + 77.2T + 4.48e3T^{2} \) |
| 71 | \( 1 + (24.2 - 33.3i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (17.5 - 53.9i)T + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-41.1 + 29.9i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-34.8 + 47.9i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 38.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (13.1 - 9.57i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−16.58283659521119381449381366870, −15.40291522277945268538438742661, −14.64267772528821294190163644679, −13.51796274285389896073482939925, −11.53545142799546268873279115907, −10.54494854504376469276125266778, −9.136280680107152238226796630037, −7.01994864841927805147273405385, −5.60365282918432974170557091910, −4.18421250827964618658076547380,
2.39612955760726099805731489055, 4.85907748114585960391054737003, 6.93049251528552613569328596988, 8.221198716922843398001639800477, 10.37703645895375578660290195519, 11.96415461959430507618276781706, 12.30956372398410208476765901368, 13.49080640963796320523390559878, 14.82811707493316220643979480387, 16.60052660216484665093726897260
