| L(s) = 1 | + (−0.533 + 2.33i)2-s + (1.50 + 0.856i)3-s + (−3.37 − 1.62i)4-s + (0.810 + 2.06i)5-s + (−2.80 + 3.06i)6-s + (1.78 − 1.95i)7-s + (2.60 − 3.26i)8-s + (1.53 + 2.57i)9-s + (−5.25 + 0.792i)10-s + (−3.87 + 1.19i)11-s + (−3.68 − 5.33i)12-s + (−0.720 + 0.222i)13-s + (3.60 + 5.21i)14-s + (−0.547 + 3.80i)15-s + (1.57 + 1.96i)16-s + (−0.536 + 7.16i)17-s + ⋯ |
| L(s) = 1 | + (−0.377 + 1.65i)2-s + (0.869 + 0.494i)3-s + (−1.68 − 0.811i)4-s + (0.362 + 0.923i)5-s + (−1.14 + 1.24i)6-s + (0.675 − 0.737i)7-s + (0.920 − 1.15i)8-s + (0.511 + 0.859i)9-s + (−1.66 + 0.250i)10-s + (−1.16 + 0.360i)11-s + (−1.06 − 1.53i)12-s + (−0.199 + 0.0616i)13-s + (0.963 + 1.39i)14-s + (−0.141 + 0.981i)15-s + (0.392 + 0.492i)16-s + (−0.130 + 1.73i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.130474 - 1.42640i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.130474 - 1.42640i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.50 - 0.856i)T \) |
| 7 | \( 1 + (-1.78 + 1.95i)T \) |
| good | 2 | \( 1 + (0.533 - 2.33i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (-0.810 - 2.06i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (3.87 - 1.19i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (0.720 - 0.222i)T + (10.7 - 7.32i)T^{2} \) |
| 17 | \( 1 + (0.536 - 7.16i)T + (-16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (0.480 + 0.832i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.508 - 0.346i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.599 + 8.00i)T + (-28.6 - 4.32i)T^{2} \) |
| 31 | \( 1 - 0.283T + 31T^{2} \) |
| 37 | \( 1 + (-9.54 - 6.50i)T + (13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-3.34 - 0.504i)T + (39.1 + 12.0i)T^{2} \) |
| 43 | \( 1 + (5.57 - 0.840i)T + (41.0 - 12.6i)T^{2} \) |
| 47 | \( 1 + (-0.605 + 2.65i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-11.0 + 7.51i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (3.71 + 4.66i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-7.61 + 3.66i)T + (38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 0.902T + 67T^{2} \) |
| 71 | \( 1 + (0.131 + 0.0631i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-6.50 - 2.00i)T + (60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + 1.44T + 79T^{2} \) |
| 83 | \( 1 + (-0.715 - 0.220i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (10.1 - 9.40i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (-5.42 + 9.40i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.13764033150323194728738006574, −10.19012639507374921006102327816, −9.830186873367269486671855639003, −8.331501261546436699166719627255, −8.046675055485958173323920077749, −7.15849196594799003133422525001, −6.22744452067276538697207218209, −5.02114240387591900857116568833, −4.06718255676453934303196406715, −2.36861352769018136483147758122,
0.981610045312689323748413221722, 2.27403424416321533059172825294, 2.94785742600732109934751845356, 4.51779973692956709093360470231, 5.48186858330998034779159824674, 7.41839850283486745148464532522, 8.406000873948940480329578291480, 9.012132350401731492034355642043, 9.592001252003474074865628336073, 10.68805434770750345584137913981
