| L(s) = 1 | + (1.81 + 1.19i)2-s + (1.08 + 2.51i)4-s + (0.443 + 0.470i)5-s + (1.81 − 0.212i)7-s + (−0.277 + 1.57i)8-s + (0.244 + 1.38i)10-s + (−0.346 − 1.15i)11-s + (0.310 + 5.32i)13-s + (3.55 + 1.78i)14-s + (1.36 − 1.44i)16-s + (6.43 + 2.34i)17-s + (−5.97 + 2.17i)19-s + (−0.700 + 1.62i)20-s + (0.754 − 2.52i)22-s + (−3.09 − 0.361i)23-s + ⋯ |
| L(s) = 1 | + (1.28 + 0.845i)2-s + (0.541 + 1.25i)4-s + (0.198 + 0.210i)5-s + (0.686 − 0.0802i)7-s + (−0.0980 + 0.556i)8-s + (0.0772 + 0.438i)10-s + (−0.104 − 0.349i)11-s + (0.0860 + 1.47i)13-s + (0.950 + 0.477i)14-s + (0.341 − 0.362i)16-s + (1.56 + 0.567i)17-s + (−1.37 + 0.499i)19-s + (−0.156 + 0.362i)20-s + (0.160 − 0.537i)22-s + (−0.645 − 0.0754i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.54262 + 2.11097i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.54262 + 2.11097i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.81 - 1.19i)T + (0.792 + 1.83i)T^{2} \) |
| 5 | \( 1 + (-0.443 - 0.470i)T + (-0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-1.81 + 0.212i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (0.346 + 1.15i)T + (-9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (-0.310 - 5.32i)T + (-12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (-6.43 - 2.34i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (5.97 - 2.17i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (3.09 + 0.361i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (-5.26 + 2.64i)T + (17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (1.65 - 2.22i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (1.09 - 0.918i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-0.931 + 0.612i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (9.37 + 2.22i)T + (38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (3.64 + 4.89i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (4.26 + 7.38i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.598 + 2.00i)T + (-49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (1.42 - 3.29i)T + (-41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (-1.09 - 0.547i)T + (40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (1.41 + 8.02i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.11 + 6.32i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-11.9 - 7.83i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-5.61 - 3.69i)T + (32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (-2.70 + 15.3i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.55 + 2.71i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.61173372994392213053094582979, −9.855823418734681897801442298795, −8.432042154286008256544187758910, −7.898862227127056008563295711260, −6.62872539133766284244060086673, −6.24810431545704361116498236423, −5.15768260696644565900419153635, −4.32543014872913909634100125818, −3.47225661467195603475911164803, −1.88205563433814724578555949218,
1.39823534481989149961864900252, 2.66570551490922819766000504494, 3.58021716705812644144673941140, 4.82115064737566443814505245761, 5.28659527579703013200083374138, 6.24328814473212777184784633700, 7.69569505770947564239668329070, 8.369577320630436448763509215967, 9.717680710271106853356932432887, 10.50679636069728375872607700571
