| L(s) = 1 | + (−0.168 − 0.156i)2-s + (−0.145 − 1.94i)4-s + (−0.711 − 1.81i)5-s + (−2.04 + 1.68i)7-s + (−0.564 + 0.708i)8-s + (−0.163 + 0.415i)10-s + (1.25 − 0.387i)11-s + (0.866 − 3.79i)13-s + (0.606 + 0.0357i)14-s + (−3.64 + 0.549i)16-s + (−5.18 − 3.53i)17-s + (−1.89 + 3.28i)19-s + (−3.41 + 1.64i)20-s + (−0.272 − 0.131i)22-s + (−3.21 + 2.19i)23-s + ⋯ |
| L(s) = 1 | + (−0.118 − 0.110i)2-s + (−0.0727 − 0.970i)4-s + (−0.318 − 0.810i)5-s + (−0.771 + 0.635i)7-s + (−0.199 + 0.250i)8-s + (−0.0516 + 0.131i)10-s + (0.379 − 0.116i)11-s + (0.240 − 1.05i)13-s + (0.162 + 0.00954i)14-s + (−0.911 + 0.137i)16-s + (−1.25 − 0.857i)17-s + (−0.435 + 0.754i)19-s + (−0.763 + 0.367i)20-s + (−0.0580 − 0.0279i)22-s + (−0.671 + 0.457i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0929904 - 0.640506i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0929904 - 0.640506i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (2.04 - 1.68i)T \) |
| good | 2 | \( 1 + (0.168 + 0.156i)T + (0.149 + 1.99i)T^{2} \) |
| 5 | \( 1 + (0.711 + 1.81i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-1.25 + 0.387i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 3.79i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (5.18 + 3.53i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (1.89 - 3.28i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.21 - 2.19i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (5.98 - 2.88i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (0.842 + 1.45i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0956 - 1.27i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-5.27 + 6.61i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (3.86 + 4.84i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-2.40 - 2.23i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (0.377 + 5.03i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-4.20 + 10.7i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.0815 + 1.08i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-4.35 - 7.53i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-13.6 - 6.58i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-9.63 + 8.94i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (-2.17 + 3.77i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.19 + 9.63i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-4.32 - 1.33i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−10.67945264732499693644023534522, −9.678434723034947528082423805482, −9.044583248429328961746916173669, −8.254937973368024235000379347836, −6.81305057252758602472822151066, −5.83531192022370734698747879418, −5.09428656008123971541731992490, −3.76845611281145665238615895504, −2.14388392205053635399449660530, −0.40379288654066264521260823133,
2.43850901645344471837973163121, 3.75467303848528815899589521321, 4.29276847280246917889578537653, 6.48537642548237318124539760440, 6.76130948713969227003308020260, 7.76659897731115554162141990521, 8.841783429240504285672600736744, 9.578593690680494911270654314882, 10.87947127987442615678394742420, 11.32953867196394688138180789710
