| L(s) = 1 | + (−0.309 + 0.951i)2-s + (−2.31 − 1.68i)3-s + (−0.809 − 0.587i)4-s + (−0.771 − 2.09i)5-s + (2.31 − 1.68i)6-s + 7-s + (0.809 − 0.587i)8-s + (1.59 + 4.92i)9-s + (2.23 − 0.0847i)10-s + (−0.391 + 1.20i)11-s + (0.883 + 2.71i)12-s + (−1.70 − 5.25i)13-s + (−0.309 + 0.951i)14-s + (−1.74 + 6.15i)15-s + (0.309 + 0.951i)16-s + (−3.52 + 2.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−1.33 − 0.970i)3-s + (−0.404 − 0.293i)4-s + (−0.344 − 0.938i)5-s + (0.944 − 0.686i)6-s + 0.377·7-s + (0.286 − 0.207i)8-s + (0.532 + 1.64i)9-s + (0.706 − 0.0267i)10-s + (−0.118 + 0.363i)11-s + (0.255 + 0.784i)12-s + (−0.473 − 1.45i)13-s + (−0.0825 + 0.254i)14-s + (−0.450 + 1.58i)15-s + (0.0772 + 0.237i)16-s + (−0.855 + 0.621i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0100827 + 0.0664447i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0100827 + 0.0664447i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.771 + 2.09i)T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + (2.31 + 1.68i)T + (0.927 + 2.85i)T^{2} \) |
| 11 | \( 1 + (0.391 - 1.20i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.70 + 5.25i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (3.52 - 2.56i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (4.97 - 3.61i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.43 - 4.40i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-4.35 - 3.16i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.00 - 1.45i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.79 + 8.61i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.287 - 0.884i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 7.47T + 43T^{2} \) |
| 47 | \( 1 + (-0.268 - 0.195i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (7.82 + 5.68i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.30 - 10.1i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.856 + 2.63i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (6.62 - 4.80i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (9.79 + 7.11i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.988 + 3.04i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (12.4 + 9.02i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (7.83 - 5.69i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.682 + 2.09i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (7.80 + 5.67i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.99956112001642779828068563443, −10.26547710517616068084841604360, −8.786130113968645942658953344097, −7.897442938543229416558235286875, −7.22954341034605554973118049464, −5.98421070868334990195259299062, −5.38878738384682268160803810064, −4.35773420395209077599777303210, −1.59685600883208112691761634788, −0.05886027887802578924535177823,
2.52480262223276858950446227437, 4.22984139632458722121283485174, 4.65898038369576130678319609814, 6.22573918501559916100100055526, 6.98739871358808423556335482151, 8.557184546035658125478913209944, 9.595039738608229644185505098511, 10.45519420421597597473347607446, 11.15299175012919604902292554416, 11.51937814989769583385706346819
