| L(s) = 1 | + (0.153 − 1.45i)2-s + (1.08 + 1.20i)3-s + (−0.147 − 0.0314i)4-s + (−0.426 − 0.189i)5-s + (1.91 − 1.39i)6-s + (0.837 − 2.57i)8-s + (0.0399 − 0.379i)9-s + (−0.342 + 0.592i)10-s + (3.24 − 0.702i)11-s + (−0.122 − 0.212i)12-s + (−1.28 − 0.930i)13-s + (−0.233 − 0.718i)15-s + (−3.90 − 1.74i)16-s + (0.546 + 5.19i)17-s + (−0.547 − 0.116i)18-s + (4.12 − 0.877i)19-s + ⋯ |
| L(s) = 1 | + (0.108 − 1.03i)2-s + (0.625 + 0.694i)3-s + (−0.0739 − 0.0157i)4-s + (−0.190 − 0.0848i)5-s + (0.783 − 0.569i)6-s + (0.296 − 0.911i)8-s + (0.0133 − 0.126i)9-s + (−0.108 + 0.187i)10-s + (0.977 − 0.211i)11-s + (−0.0353 − 0.0612i)12-s + (−0.355 − 0.257i)13-s + (−0.0602 − 0.185i)15-s + (−0.977 − 0.435i)16-s + (0.132 + 1.26i)17-s + (−0.129 − 0.0274i)18-s + (0.947 − 0.201i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81382 - 1.06705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81382 - 1.06705i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (-3.24 + 0.702i)T \) |
| good | 2 | \( 1 + (-0.153 + 1.45i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (-1.08 - 1.20i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (0.426 + 0.189i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (1.28 + 0.930i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.546 - 5.19i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-4.12 + 0.877i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.902 - 1.56i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.840 - 2.58i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (1.18 - 0.526i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (1.29 - 1.44i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (-0.321 + 0.990i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + (6.25 - 1.32i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (12.0 - 5.37i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (8.41 + 1.78i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-13.9 - 6.19i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-2.33 + 4.04i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.88 - 5.72i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (13.0 + 2.76i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (0.374 - 3.56i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (13.9 - 10.1i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.45 + 7.72i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.18 + 1.58i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.64667164992128155451902788999, −9.842927747146380293403433827393, −9.251496670552122296407506186374, −8.260178955226787057984119939693, −7.11151793493234842302047237781, −6.02022917787993086739356297319, −4.43261378453370415077483420382, −3.69045730309177600967650621976, −2.89184500544145890030340855535, −1.35937732575795820611796251353,
1.71893340071186279030003002871, 2.97770282871874896792048401798, 4.56101286764610113782203849792, 5.59784831444353253201883209070, 6.75104078338398486534529330732, 7.33197887009475154545609924733, 7.915219044586834029560946677962, 8.950565301748585474523537843038, 9.794056874100941582558851060856, 11.22962041758854396295852012324
