| L(s) = 1 | + (1.76 + 1.16i)2-s + (0.980 + 2.27i)4-s + (−2.67 − 2.83i)5-s + (3.31 − 0.387i)7-s + (−0.175 + 0.992i)8-s + (−1.43 − 8.11i)10-s + (−0.217 − 0.724i)11-s + (−0.144 − 2.48i)13-s + (6.30 + 3.16i)14-s + (1.93 − 2.05i)16-s + (−0.700 − 0.255i)17-s + (4.21 − 1.53i)19-s + (3.81 − 8.85i)20-s + (0.459 − 1.53i)22-s + (2.27 + 0.265i)23-s + ⋯ |
| L(s) = 1 | + (1.24 + 0.822i)2-s + (0.490 + 1.13i)4-s + (−1.19 − 1.26i)5-s + (1.25 − 0.146i)7-s + (−0.0618 + 0.350i)8-s + (−0.452 − 2.56i)10-s + (−0.0654 − 0.218i)11-s + (−0.0400 − 0.688i)13-s + (1.68 + 0.846i)14-s + (0.483 − 0.512i)16-s + (−0.170 − 0.0618i)17-s + (0.967 − 0.352i)19-s + (0.853 − 1.97i)20-s + (0.0978 − 0.327i)22-s + (0.473 + 0.0553i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.69138 - 0.189398i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.69138 - 0.189398i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.76 - 1.16i)T + (0.792 + 1.83i)T^{2} \) |
| 5 | \( 1 + (2.67 + 2.83i)T + (-0.290 + 4.99i)T^{2} \) |
| 7 | \( 1 + (-3.31 + 0.387i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (0.217 + 0.724i)T + (-9.19 + 6.04i)T^{2} \) |
| 13 | \( 1 + (0.144 + 2.48i)T + (-12.9 + 1.50i)T^{2} \) |
| 17 | \( 1 + (0.700 + 0.255i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.21 + 1.53i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.27 - 0.265i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (-0.414 + 0.208i)T + (17.3 - 23.2i)T^{2} \) |
| 31 | \( 1 + (2.35 - 3.16i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (3.64 - 3.05i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (4.08 - 2.68i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (-5.78 - 1.37i)T + (38.4 + 19.2i)T^{2} \) |
| 47 | \( 1 + (6.42 + 8.63i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (-5.75 - 9.96i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.19 + 3.97i)T + (-49.2 - 32.4i)T^{2} \) |
| 61 | \( 1 + (-0.105 + 0.245i)T + (-41.8 - 44.3i)T^{2} \) |
| 67 | \( 1 + (1.71 + 0.860i)T + (40.0 + 53.7i)T^{2} \) |
| 71 | \( 1 + (-1.17 - 6.65i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.37 + 7.81i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (3.60 + 2.37i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-2.12 - 1.39i)T + (32.8 + 76.2i)T^{2} \) |
| 89 | \( 1 + (0.935 - 5.30i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (6.51 - 6.90i)T + (-5.64 - 96.8i)T^{2} \) |
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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
−10.64707004246404948579695207231, −9.191491486883627400004247649641, −8.218125100630280113540718045620, −7.76845772341734487826361691059, −6.91279236274263225698239756198, −5.37420847997605126946108109576, −5.05647489098716021003075791064, −4.25870519570237633833012893185, −3.29973350769793388944727429144, −1.05670405618300934165455479208,
1.86920244061635275062714649492, 2.98298553983291006803537460867, 3.89169021805219354157770871715, 4.62567949285585643906946584060, 5.62541029202368407876579015838, 6.91771195959969648194450919071, 7.66118942937819561959256193576, 8.536768790234381878680664663512, 10.00927070077057272281127873217, 11.04538370132546170863965743396
