| L(s) = 1 | + (0.223 − 1.26i)2-s + (0.326 + 0.118i)4-s + (0.342 − 0.286i)5-s + (3.31 − 1.20i)7-s + (1.50 − 2.61i)8-s + (−0.286 − 0.497i)10-s + (−2.12 − 1.78i)11-s + (0.571 + 3.24i)13-s + (−0.788 − 4.47i)14-s + (−2.43 − 2.04i)16-s + (3.51 + 6.09i)17-s + (2.59 − 4.49i)19-s + (0.145 − 0.0530i)20-s + (−2.73 + 2.29i)22-s + (−6.83 − 2.48i)23-s + ⋯ |
| L(s) = 1 | + (0.157 − 0.895i)2-s + (0.163 + 0.0593i)4-s + (0.152 − 0.128i)5-s + (1.25 − 0.456i)7-s + (0.533 − 0.923i)8-s + (−0.0907 − 0.157i)10-s + (−0.642 − 0.538i)11-s + (0.158 + 0.898i)13-s + (−0.210 − 1.19i)14-s + (−0.609 − 0.511i)16-s + (0.853 + 1.47i)17-s + (0.594 − 1.03i)19-s + (0.0325 − 0.0118i)20-s + (−0.583 + 0.489i)22-s + (−1.42 − 0.518i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64048 - 1.45990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64048 - 1.45990i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.223 + 1.26i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-0.342 + 0.286i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.31 + 1.20i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.12 + 1.78i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.571 - 3.24i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.51 - 6.09i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.59 + 4.49i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.83 + 2.48i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.628 + 3.56i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.81 - 0.662i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.61 - 2.79i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.844 + 4.78i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (4.41 + 3.70i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.83 - 1.03i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 8.77T + 53T^{2} \) |
| 59 | \( 1 + (-2.27 + 1.90i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (7.41 - 2.69i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (1.63 + 9.29i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.65 + 4.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.777 + 1.34i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.06 - 11.7i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.82 - 16.0i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (9.21 - 15.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.74 - 6.50i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.48765668331469789785800989253, −9.628046613860613007356083133450, −8.360261654486597762108794806131, −7.77682128830913304920251701123, −6.71966552652010550501180642994, −5.55987677790874399985789482485, −4.43779963788571805449049321832, −3.60440898636823535859337145282, −2.23383950060896805318108107847, −1.28031084662150670435868966997,
1.68886000283347128355313421753, 2.87006991808792782396662685596, 4.62212709714873164596355649478, 5.41972235232899322299572596489, 5.94706746313735144991734371450, 7.33367556318084413842246288322, 7.82476387687152806052289503713, 8.429559932289503579359023539947, 9.903594581397341581622148746310, 10.42204284882412835811045848123
