| L(s) = 1 | + (0.0871 + 0.996i)2-s + (−1.71 − 0.271i)3-s + (−0.984 + 0.173i)4-s + (−1.70 − 1.44i)5-s + (0.121 − 1.72i)6-s + (−3.51 − 0.941i)7-s + (−0.258 − 0.965i)8-s + (2.85 + 0.929i)9-s + (1.28 − 1.82i)10-s + (1.29 + 0.747i)11-s + (1.73 − 0.0295i)12-s + (1.36 + 2.93i)13-s + (0.631 − 3.58i)14-s + (2.53 + 2.93i)15-s + (0.939 − 0.342i)16-s + (−0.302 − 3.45i)17-s + ⋯ |
| L(s) = 1 | + (0.0616 + 0.704i)2-s + (−0.987 − 0.156i)3-s + (−0.492 + 0.0868i)4-s + (−0.764 − 0.644i)5-s + (0.0495 − 0.705i)6-s + (−1.32 − 0.355i)7-s + (−0.0915 − 0.341i)8-s + (0.950 + 0.309i)9-s + (0.407 − 0.578i)10-s + (0.390 + 0.225i)11-s + (0.499 − 0.00853i)12-s + (0.379 + 0.814i)13-s + (0.168 − 0.957i)14-s + (0.653 + 0.756i)15-s + (0.234 − 0.0855i)16-s + (−0.0733 − 0.838i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.594332 + 0.352184i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.594332 + 0.352184i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0871 - 0.996i)T \) |
| 3 | \( 1 + (1.71 + 0.271i)T \) |
| 5 | \( 1 + (1.70 + 1.44i)T \) |
| 19 | \( 1 + (-2.58 - 3.51i)T \) |
| good | 7 | \( 1 + (3.51 + 0.941i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.29 - 0.747i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.36 - 2.93i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (0.302 + 3.45i)T + (-16.7 + 2.95i)T^{2} \) |
| 23 | \( 1 + (-3.19 - 2.23i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-3.57 + 3.00i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.43 - 7.68i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.15 + 5.15i)T - 37iT^{2} \) |
| 41 | \( 1 + (1.28 + 3.53i)T + (-31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (2.95 - 2.07i)T + (14.7 - 40.4i)T^{2} \) |
| 47 | \( 1 + (5.46 + 0.478i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (-8.38 - 5.87i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (7.15 + 6.00i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.00426 + 0.0241i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.0962 - 1.09i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-5.99 - 1.05i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-6.61 - 3.08i)T + (46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (-5.00 - 13.7i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.3 - 3.30i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-9.69 - 3.52i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.56 - 0.137i)T + (95.5 - 16.8i)T^{2} \) |
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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
−10.98840287979287896456600900506, −9.796664081296041781485496480488, −9.248942668472317868338905332067, −8.010337952835239086344761459972, −6.99466186595923353188647997061, −6.56091800136651645479302744322, −5.41393232835300170195636900671, −4.45943455481940523314131576816, −3.53278954167405405516686433000, −0.925844025486922012423139857916,
0.63736973490810053393453413707, 2.90229387048595667942772282989, 3.69177976380563216644102578074, 4.84513223110853264804519008760, 6.16666089155836666925185576293, 6.62850890073109434821110098700, 7.940368304083311119898687195305, 9.119207978479393347357751966380, 10.05656234424991024320923936004, 10.63618197356742630145332042399
