| L(s) = 1 | + (−0.0299 + 1.41i)2-s + (1.06 + 0.770i)3-s + (−1.99 − 0.0847i)4-s + (−1.15 + 1.91i)5-s + (−1.12 + 1.47i)6-s + 2.31i·7-s + (0.179 − 2.82i)8-s + (−0.395 − 1.21i)9-s + (−2.67 − 1.69i)10-s + (−0.219 − 0.0714i)11-s + (−2.05 − 1.63i)12-s + (0.659 + 2.02i)13-s + (−3.27 − 0.0693i)14-s + (−2.70 + 1.13i)15-s + (3.98 + 0.338i)16-s + (1.33 + 1.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.0211 + 0.999i)2-s + (0.612 + 0.445i)3-s + (−0.999 − 0.0423i)4-s + (−0.517 + 0.855i)5-s + (−0.458 + 0.603i)6-s + 0.875i·7-s + (0.0635 − 0.997i)8-s + (−0.131 − 0.405i)9-s + (−0.844 − 0.535i)10-s + (−0.0662 − 0.0215i)11-s + (−0.593 − 0.470i)12-s + (0.182 + 0.562i)13-s + (−0.875 − 0.0185i)14-s + (−0.697 + 0.294i)15-s + (0.996 + 0.0846i)16-s + (0.324 + 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 - 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.833 - 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.328248 + 1.09050i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.328248 + 1.09050i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0299 - 1.41i)T \) |
| 5 | \( 1 + (1.15 - 1.91i)T \) |
| good | 3 | \( 1 + (-1.06 - 0.770i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 - 2.31iT - 7T^{2} \) |
| 11 | \( 1 + (0.219 + 0.0714i)T + (8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-0.659 - 2.02i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.33 - 1.84i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.307 + 0.423i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-6.59 - 2.14i)T + (18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (5.13 - 7.06i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.95 + 5.05i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.46 + 7.58i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.84 - 5.67i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 9.74T + 43T^{2} \) |
| 47 | \( 1 + (0.657 - 0.904i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (3.51 + 2.55i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.66 + 1.51i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.16 + 0.704i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-2.55 + 1.85i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (5.30 + 3.85i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.24 + 1.70i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (10.1 + 7.39i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.44 + 3.23i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.88 - 15.0i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-7.50 + 10.3i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−13.05339083775015995980046876122, −11.95800897785618961875116055800, −10.78086989976233325354162983872, −9.466052143469782008045324963419, −8.877423425402765657619941964368, −7.81940036583751256855731911048, −6.74480693514118012574942135973, −5.70330334881080378660378590212, −4.17223455667388708000715119086, −3.05437197982431464329906346305,
1.05935157979115124125677371626, 2.85597659147152071453658968923, 4.18564953799095653049604176163, 5.27387174915707692943093764990, 7.39249708322829599071208343309, 8.201254249350894281237021335462, 9.022481802702736388709589486888, 10.22773255097099464207539690562, 11.12667809785733793243747267311, 12.15167668856824179934970553020
