| L(s) = 1 | + (−0.984 − 0.173i)2-s + (1.28 − 1.15i)3-s + (0.939 + 0.342i)4-s + (−1.01 − 1.99i)5-s + (−1.46 + 0.917i)6-s + (3.34 − 1.93i)7-s + (−0.866 − 0.5i)8-s + (0.313 − 2.98i)9-s + (0.654 + 2.13i)10-s + (−3.47 − 2.00i)11-s + (1.60 − 0.648i)12-s + (0.681 − 0.571i)13-s + (−3.63 + 1.32i)14-s + (−3.61 − 1.38i)15-s + (0.766 + 0.642i)16-s + (−0.566 + 3.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 − 0.122i)2-s + (0.743 − 0.669i)3-s + (0.469 + 0.171i)4-s + (−0.454 − 0.890i)5-s + (−0.599 + 0.374i)6-s + (1.26 − 0.730i)7-s + (−0.306 − 0.176i)8-s + (0.104 − 0.994i)9-s + (0.207 + 0.676i)10-s + (−1.04 − 0.605i)11-s + (0.463 − 0.187i)12-s + (0.189 − 0.158i)13-s + (−0.970 + 0.353i)14-s + (−0.933 − 0.357i)15-s + (0.191 + 0.160i)16-s + (−0.137 + 0.778i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.596062 - 1.12248i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.596062 - 1.12248i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (-1.28 + 1.15i)T \) |
| 5 | \( 1 + (1.01 + 1.99i)T \) |
| 19 | \( 1 + (3.42 - 2.69i)T \) |
| good | 7 | \( 1 + (-3.34 + 1.93i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3.47 + 2.00i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.681 + 0.571i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.566 - 3.21i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-6.82 - 2.48i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.08 + 6.16i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.50 + 0.868i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.85T + 37T^{2} \) |
| 41 | \( 1 + (-4.43 - 3.72i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.819 + 2.25i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.50 - 8.51i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.88 + 10.6i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.00803 + 0.0455i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.60 + 0.946i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.646 + 3.66i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (11.9 - 4.35i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-7.68 + 9.15i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.79 + 9.29i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.50 - 2.60i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.50 + 5.45i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (2.39 - 13.5i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.52040706337051734639651709886, −9.282647670649059316318594799756, −8.350789790013244632220509550248, −8.041564706775479997182845117476, −7.38018881457401443910950629770, −5.98469636745230852985334787032, −4.66072525894184634690284523584, −3.51242055507666379798242066472, −1.96871174561116455734282016185, −0.852094383287629326746468473373,
2.17967180490362007088015725377, 2.92115039733379832868420036303, 4.53053928481329775201928061756, 5.32618888695911737627688914069, 6.98792291612790922400333565650, 7.60819431959973344480996907648, 8.559727358307927657837391524098, 9.031151053309756524424477100308, 10.29014513796340905290388346864, 10.84118114669357746862377972762
