| L(s) = 1 | + (−1.34 − 0.100i)2-s + (0.896 + 1.48i)3-s + (−0.178 − 0.0268i)4-s + (−0.507 − 0.471i)5-s + (−1.05 − 2.08i)6-s + (1.99 + 1.73i)7-s + (2.86 + 0.654i)8-s + (−1.39 + 2.65i)9-s + (0.635 + 0.685i)10-s + (−2.98 + 4.37i)11-s + (−0.119 − 0.288i)12-s + (−0.223 − 0.463i)13-s + (−2.50 − 2.53i)14-s + (0.243 − 1.17i)15-s + (−3.44 − 1.06i)16-s + (−1.27 + 3.25i)17-s + ⋯ |
| L(s) = 1 | + (−0.951 − 0.0712i)2-s + (0.517 + 0.855i)3-s + (−0.0891 − 0.0134i)4-s + (−0.227 − 0.210i)5-s + (−0.431 − 0.850i)6-s + (0.753 + 0.656i)7-s + (1.01 + 0.231i)8-s + (−0.464 + 0.885i)9-s + (0.200 + 0.216i)10-s + (−0.899 + 1.31i)11-s + (−0.0346 − 0.0832i)12-s + (−0.0619 − 0.128i)13-s + (−0.670 − 0.678i)14-s + (0.0627 − 0.303i)15-s + (−0.861 − 0.265i)16-s + (−0.309 + 0.788i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.538078 + 0.472009i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.538078 + 0.472009i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.896 - 1.48i)T \) |
| 7 | \( 1 + (-1.99 - 1.73i)T \) |
| good | 2 | \( 1 + (1.34 + 0.100i)T + (1.97 + 0.298i)T^{2} \) |
| 5 | \( 1 + (0.507 + 0.471i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (2.98 - 4.37i)T + (-4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (0.223 + 0.463i)T + (-8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (1.27 - 3.25i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (-2.46 - 1.42i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.23 + 2.83i)T + (16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-1.66 + 1.33i)T + (6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.27 + 1.88i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.78 - 0.420i)T + (35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (-2.45 + 10.7i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (1.18 + 5.17i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (0.418 - 5.58i)T + (-46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (-0.696 + 4.62i)T + (-50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-2.94 + 2.73i)T + (4.40 - 58.8i)T^{2} \) |
| 61 | \( 1 + (-2.18 - 14.5i)T + (-58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (2.58 + 4.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.79 + 3.02i)T + (15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (2.80 - 0.210i)T + (72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-2.78 + 4.81i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.70 - 4.67i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (-8.16 + 5.56i)T + (32.5 - 82.8i)T^{2} \) |
| 97 | \( 1 - 7.70iT - 97T^{2} \) |
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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
−13.34359575182054439393595759670, −12.15349970935362425063010653723, −10.75752656121571724881842240518, −10.17104393034088898830173307800, −9.078879319407605230258411867231, −8.363787628450853613608092539237, −7.52033375990702425244210161865, −5.20253266915447991034486463059, −4.40853490892561429282945257710, −2.26552511456550769553680862746,
1.01642994984873305491913694994, 3.16010693823882170031269261727, 5.04637096798081888968151850876, 6.95719912724899131746772748381, 7.73208428207652470089227530960, 8.494063760747531525739015429565, 9.449903024463543942302851897199, 10.85465448893876270848813557775, 11.50897020369269355633381677298, 13.28380452234083431149365764405
