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
−10.09990343446598210834014401683, −9.648710204000159769259319606555, −9.395694743100435992384329695416, −8.343094860463593164308555409060, −7.05129492376354082013803296778, −6.67533604932663054826781498118, −5.61313940240864155344899868788, −3.60450779205398073054856602242, −2.34378502146102025051138785158, −1.58866920609141876564444562866,
0.39760602714922788967132659332, 2.05786687189515553081331843359, 2.92483133277921069392842952590, 5.42080466313018767746418172601, 6.18255967782387698140102995046, 6.63768957718905710245302529399, 7.67534581166627070110748371327, 8.867407918918050315214116761140, 9.276060060057763129788687699862, 9.953797932578669069832526313943