| L(s) = 1 | + (−2.57 + 0.718i)2-s + (4.42 − 2.67i)4-s + (3.51 + 0.136i)5-s + (−3.43 + 0.536i)7-s + (−5.82 + 6.17i)8-s + (−9.17 + 2.17i)10-s + (0.305 − 2.23i)11-s + (−1.63 − 2.37i)13-s + (8.46 − 3.84i)14-s + (5.77 − 10.9i)16-s + (−1.77 − 0.892i)17-s + (0.153 − 2.62i)19-s + (15.9 − 8.79i)20-s + (0.817 + 5.98i)22-s + (−1.99 − 5.15i)23-s + ⋯ |
| L(s) = 1 | + (−1.82 + 0.507i)2-s + (2.21 − 1.33i)4-s + (1.57 + 0.0610i)5-s + (−1.29 + 0.202i)7-s + (−2.06 + 2.18i)8-s + (−2.89 + 0.687i)10-s + (0.0920 − 0.673i)11-s + (−0.452 − 0.659i)13-s + (2.26 − 1.02i)14-s + (1.44 − 2.74i)16-s + (−0.430 − 0.216i)17-s + (0.0351 − 0.603i)19-s + (3.56 − 1.96i)20-s + (0.174 + 1.27i)22-s + (−0.415 − 1.07i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0989 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0989 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.340491 - 0.308327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.340491 - 0.308327i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (2.57 - 0.718i)T + (1.71 - 1.03i)T^{2} \) |
| 5 | \( 1 + (-3.51 - 0.136i)T + (4.98 + 0.387i)T^{2} \) |
| 7 | \( 1 + (3.43 - 0.536i)T + (6.66 - 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.305 + 2.23i)T + (-10.5 - 2.94i)T^{2} \) |
| 13 | \( 1 + (1.63 + 2.37i)T + (-4.68 + 12.1i)T^{2} \) |
| 17 | \( 1 + (1.77 + 0.892i)T + (10.1 + 13.6i)T^{2} \) |
| 19 | \( 1 + (-0.153 + 2.62i)T + (-18.8 - 2.20i)T^{2} \) |
| 23 | \( 1 + (1.99 + 5.15i)T + (-17.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (3.05 - 2.18i)T + (9.38 - 27.4i)T^{2} \) |
| 31 | \( 1 + (5.26 + 6.03i)T + (-4.19 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-0.347 + 0.466i)T + (-10.6 - 35.4i)T^{2} \) |
| 41 | \( 1 + (0.455 + 1.77i)T + (-35.8 + 19.8i)T^{2} \) |
| 43 | \( 1 + (-0.802 + 0.728i)T + (4.16 - 42.7i)T^{2} \) |
| 47 | \( 1 + (7.33 - 8.39i)T + (-6.36 - 46.5i)T^{2} \) |
| 53 | \( 1 + (0.570 + 3.23i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-10.5 - 4.29i)T + (42.1 + 41.3i)T^{2} \) |
| 61 | \( 1 + (-2.86 - 1.72i)T + (28.4 + 53.9i)T^{2} \) |
| 67 | \( 1 + (5.67 + 4.05i)T + (21.6 + 63.3i)T^{2} \) |
| 71 | \( 1 + (-3.36 + 11.2i)T + (-59.3 - 39.0i)T^{2} \) |
| 73 | \( 1 + (-5.59 - 1.32i)T + (65.2 + 32.7i)T^{2} \) |
| 79 | \( 1 + (1.57 + 1.54i)T + (1.53 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-3.97 + 15.4i)T + (-72.6 - 40.0i)T^{2} \) |
| 89 | \( 1 + (2.46 + 8.23i)T + (-74.3 + 48.9i)T^{2} \) |
| 97 | \( 1 + (-4.01 + 0.155i)T + (96.7 - 7.51i)T^{2} \) |
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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
−9.953797932578669069832526313943, −9.276060060057763129788687699862, −8.867407918918050315214116761140, −7.67534581166627070110748371327, −6.63768957718905710245302529399, −6.18255967782387698140102995046, −5.42080466313018767746418172601, −2.92483133277921069392842952590, −2.05786687189515553081331843359, −0.39760602714922788967132659332,
1.58866920609141876564444562866, 2.34378502146102025051138785158, 3.60450779205398073054856602242, 5.61313940240864155344899868788, 6.67533604932663054826781498118, 7.05129492376354082013803296778, 8.343094860463593164308555409060, 9.395694743100435992384329695416, 9.648710204000159769259319606555, 10.09990343446598210834014401683
