| L(s) = 1 | + (−0.416 + 1.82i)2-s + (−0.611 + 1.62i)3-s + (−1.35 − 0.650i)4-s + (−0.0698 − 0.177i)5-s + (−2.70 − 1.78i)6-s + (2.35 + 1.21i)7-s + (−0.583 + 0.731i)8-s + (−2.25 − 1.98i)9-s + (0.353 − 0.0533i)10-s + (−2.48 + 0.766i)11-s + (1.88 − 1.79i)12-s + (−4.36 + 1.34i)13-s + (−3.18 + 3.78i)14-s + (0.331 − 0.00444i)15-s + (−2.96 − 3.71i)16-s + (0.0993 − 1.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.294 + 1.28i)2-s + (−0.352 + 0.935i)3-s + (−0.675 − 0.325i)4-s + (−0.0312 − 0.0795i)5-s + (−1.10 − 0.730i)6-s + (0.888 + 0.458i)7-s + (−0.206 + 0.258i)8-s + (−0.751 − 0.660i)9-s + (0.111 − 0.0168i)10-s + (−0.749 + 0.231i)11-s + (0.542 − 0.517i)12-s + (−1.21 + 0.373i)13-s + (−0.852 + 1.01i)14-s + (0.0854 − 0.00114i)15-s + (−0.740 − 0.928i)16-s + (0.0240 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.517 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.362033 - 0.641789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.362033 - 0.641789i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.611 - 1.62i)T \) |
| 7 | \( 1 + (-2.35 - 1.21i)T \) |
| good | 2 | \( 1 + (0.416 - 1.82i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (0.0698 + 0.177i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (2.48 - 0.766i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (4.36 - 1.34i)T + (10.7 - 7.32i)T^{2} \) |
| 17 | \( 1 + (-0.0993 + 1.32i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-1.53 - 2.65i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.43 - 3.70i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (0.210 - 2.80i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 - 0.744T + 31T^{2} \) |
| 37 | \( 1 + (-1.54 - 1.05i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (7.29 + 1.09i)T + (39.1 + 12.0i)T^{2} \) |
| 43 | \( 1 + (-12.8 + 1.94i)T + (41.0 - 12.6i)T^{2} \) |
| 47 | \( 1 + (2.22 - 9.76i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-9.50 + 6.47i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-5.40 - 6.78i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (11.3 - 5.47i)T + (38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 2.57T + 67T^{2} \) |
| 71 | \( 1 + (6.29 + 3.02i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (1.11 + 0.345i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + 7.54T + 79T^{2} \) |
| 83 | \( 1 + (-8.82 - 2.72i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (11.1 - 10.3i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (5.58 - 9.67i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.79032780728348802257266517936, −10.63864008261665993344837499869, −9.719549346861727581970966130729, −8.886759250305149338480970406519, −7.992209454557659498055705656572, −7.23404807361489978840978166492, −5.91539925894632069951785261267, −5.23603704752870333108975376242, −4.45610931367394578340139409586, −2.59609957008516851854210635597,
0.50869621009492735143763650643, 1.95603812008663418100018369983, 2.87699903419676894791388118814, 4.52946899096535544207862717486, 5.67008388124826683739843177110, 7.00758652196786530831669803123, 7.80889639735007575596862152324, 8.731377163062681428280670686093, 10.07280504532111676193345318435, 10.66822601575686880031167334880
