| 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
−10.50679636069728375872607700571, −9.717680710271106853356932432887, −8.369577320630436448763509215967, −7.69569505770947564239668329070, −6.24328814473212777184784633700, −5.28659527579703013200083374138, −4.82115064737566443814505245761, −3.58021716705812644144673941140, −2.66570551490922819766000504494, −1.39823534481989149961864900252,
1.88205563433814724578555949218, 3.47225661467195603475911164803, 4.32543014872913909634100125818, 5.15768260696644565900419153635, 6.24810431545704361116498236423, 6.62872539133766284244060086673, 7.898862227127056008563295711260, 8.432042154286008256544187758910, 9.855823418734681897801442298795, 10.61173372994392213053094582979
