| L(s) = 1 | + (0.898 − 2.46i)2-s + (−2.94 − 0.559i)3-s + (−2.22 − 1.86i)4-s + (6.15 + 1.08i)5-s + (−4.02 + 6.77i)6-s + (−6.21 + 5.21i)7-s + (2.49 − 1.43i)8-s + (8.37 + 3.29i)9-s + (8.21 − 14.2i)10-s + (−7.99 + 1.40i)11-s + (5.51 + 6.74i)12-s + (−9.12 + 3.32i)13-s + (7.29 + 20.0i)14-s + (−17.5 − 6.64i)15-s + (−3.33 − 18.8i)16-s + (−13.0 − 7.51i)17-s + ⋯ |
| L(s) = 1 | + (0.449 − 1.23i)2-s + (−0.982 − 0.186i)3-s + (−0.556 − 0.466i)4-s + (1.23 + 0.217i)5-s + (−0.671 + 1.12i)6-s + (−0.888 + 0.745i)7-s + (0.311 − 0.179i)8-s + (0.930 + 0.366i)9-s + (0.821 − 1.42i)10-s + (−0.726 + 0.128i)11-s + (0.459 + 0.562i)12-s + (−0.701 + 0.255i)13-s + (0.521 + 1.43i)14-s + (−1.16 − 0.442i)15-s + (−0.208 − 1.18i)16-s + (−0.765 − 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.775152 - 0.637713i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.775152 - 0.637713i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.94 + 0.559i)T \) |
| good | 2 | \( 1 + (-0.898 + 2.46i)T + (-3.06 - 2.57i)T^{2} \) |
| 5 | \( 1 + (-6.15 - 1.08i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (6.21 - 5.21i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (7.99 - 1.40i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (9.12 - 3.32i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (13.0 + 7.51i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-8.93 - 15.4i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-16.9 + 20.1i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-5.18 + 14.2i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-25.0 - 21.0i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (15.8 - 27.3i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-5.65 - 15.5i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (14.5 + 82.5i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-8.46 - 10.0i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 25.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (22.3 + 3.94i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-26.1 + 21.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (7.61 - 2.77i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-35.7 - 20.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-40.4 - 69.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-25.9 - 9.45i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (6.01 - 16.5i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-24.4 + 14.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-29.2 - 166. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−17.06802069820243792461737596058, −15.79968536247108076349861795713, −13.75804432046637124846555553342, −12.80244646537044499998984600892, −11.94856106335886966297186061197, −10.47849463417616944092302355298, −9.704451242731021193943742231429, −6.66926920894158807007377805724, −5.14672744087761736737375247855, −2.42347837168797075446371670189,
4.94833638875826744627300083890, 6.09246789519869896689683914492, 7.19825236362657968301279990399, 9.627958719787947305290742809801, 10.79092359632934566182116593833, 12.98889055627038125642480181715, 13.58575375491341829878617281889, 15.27511519134173194584428313270, 16.25723093984863893139055343776, 17.19135952609744429409422404208
