| L(s) = 1 | + (0.615 + 0.571i)2-s + (0.988 + 0.149i)3-s + (−0.0967 − 1.29i)4-s + (−0.568 − 1.44i)5-s + (0.523 + 0.656i)6-s + (1.56 + 2.13i)7-s + (1.72 − 2.16i)8-s + (0.955 + 0.294i)9-s + (0.477 − 1.21i)10-s + (−1.88 + 0.582i)11-s + (0.0967 − 1.29i)12-s + (−0.828 + 3.62i)13-s + (−0.252 + 2.20i)14-s + (−0.346 − 1.51i)15-s + (−0.262 + 0.0396i)16-s + (−3.40 − 2.32i)17-s + ⋯ |
| L(s) = 1 | + (0.435 + 0.403i)2-s + (0.570 + 0.0860i)3-s + (−0.0483 − 0.645i)4-s + (−0.254 − 0.647i)5-s + (0.213 + 0.268i)6-s + (0.592 + 0.805i)7-s + (0.609 − 0.764i)8-s + (0.318 + 0.0982i)9-s + (0.150 − 0.384i)10-s + (−0.569 + 0.175i)11-s + (0.0279 − 0.372i)12-s + (−0.229 + 1.00i)13-s + (−0.0674 + 0.589i)14-s + (−0.0893 − 0.391i)15-s + (−0.0657 + 0.00990i)16-s + (−0.826 − 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0289i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.55468 + 0.0224973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.55468 + 0.0224973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.988 - 0.149i)T \) |
| 7 | \( 1 + (-1.56 - 2.13i)T \) |
| good | 2 | \( 1 + (-0.615 - 0.571i)T + (0.149 + 1.99i)T^{2} \) |
| 5 | \( 1 + (0.568 + 1.44i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (1.88 - 0.582i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (0.828 - 3.62i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (3.40 + 2.32i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (1.88 - 3.26i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.44 - 1.66i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-1.94 + 0.937i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.421 - 0.730i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.772 + 10.3i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (6.50 - 8.15i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-4.94 - 6.19i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (1.22 + 1.13i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (0.368 + 4.91i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-2.99 + 7.63i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (0.197 - 2.63i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (2.13 + 3.69i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.2 - 6.35i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (3.45 - 3.20i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (-7.12 + 12.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.636 + 2.78i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-16.3 - 5.03i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−13.23596156550182419343609230374, −12.25493834647066512328856951087, −11.10115145730961327976643435115, −9.790320365114497825578213007726, −8.900630644565897633388847421534, −7.85694568936442283010828177047, −6.47388200851927430133056480241, −5.12938669415726817043675397508, −4.32220208609062308414260601351, −2.04504432775183787395879932861,
2.50466955085602364478205778012, 3.67657592666392003259047905314, 4.86015069484897934307474506712, 6.89877115866040611921279093816, 7.84656574637575766152510591130, 8.601530899177944263246184879343, 10.44398682180918784313342266058, 10.94034881689460862385537047701, 12.17194458111552271577348626200, 13.22446026837985123127883353801
