| L(s) = 1 | + (−0.243 + 1.37i)2-s + (−0.0270 + 0.00985i)3-s + (0.0365 + 0.0133i)4-s + (−2.82 + 1.02i)5-s + (−0.00700 − 0.0397i)6-s + (2.29 + 1.31i)7-s + (−1.42 + 2.47i)8-s + (−2.29 + 1.92i)9-s + (−0.730 − 4.14i)10-s + 3.91·11-s − 0.00112·12-s + (−0.480 − 2.72i)13-s + (−2.36 + 2.85i)14-s + (0.0663 − 0.0556i)15-s + (−3.00 − 2.52i)16-s + (0.912 + 0.765i)17-s + ⋯ |
| L(s) = 1 | + (−0.171 + 0.975i)2-s + (−0.0156 + 0.00568i)3-s + (0.0182 + 0.00665i)4-s + (−1.26 + 0.459i)5-s + (−0.00286 − 0.0162i)6-s + (0.868 + 0.495i)7-s + (−0.504 + 0.874i)8-s + (−0.765 + 0.642i)9-s + (−0.231 − 1.31i)10-s + 1.18·11-s − 0.000323·12-s + (−0.133 − 0.755i)13-s + (−0.632 + 0.761i)14-s + (0.0171 − 0.0143i)15-s + (−0.750 − 0.630i)16-s + (0.221 + 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.518 - 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.460330 + 0.817224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.460330 + 0.817224i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.29 - 1.31i)T \) |
| 19 | \( 1 + (-3.07 + 3.08i)T \) |
| good | 2 | \( 1 + (0.243 - 1.37i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (0.0270 - 0.00985i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (2.82 - 1.02i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 - 3.91T + 11T^{2} \) |
| 13 | \( 1 + (0.480 + 2.72i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.912 - 0.765i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.596 + 3.38i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-8.43 - 3.06i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.256 + 0.443i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.26 + 3.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.71 + 9.71i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.37 - 5.34i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (8.23 - 6.91i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (9.59 + 3.49i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (4.55 + 3.82i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.945 - 5.36i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.504 - 2.85i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (4.59 + 3.85i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (7.06 - 2.57i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (6.81 + 5.72i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.999 - 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.61 - 3.49i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (3.94 - 1.43i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.35732631100069562309633424369, −12.28187209014516602835582828510, −11.52312935940005869114549217315, −10.89365519214445336878810683232, −8.879636634615202465058580355740, −8.078229692290638981908567791413, −7.36400012691426009919446488545, −6.08424506241129233147969804498, −4.77188432403239811482856207028, −2.91442139887807324546142689865,
1.18840999110659202600256413162, 3.41731583426919689549907412686, 4.43272922489694228810259924077, 6.41122922189014990757523027413, 7.73132472455426025118373174338, 8.848711835330885540439816480163, 9.918397272427481798993133699168, 11.47485393119143591720761121087, 11.60637926278243264876025351116, 12.28380310468910733614052098552
