| L(s) = 1 | + (−1.32 + 0.490i)2-s + (−0.941 − 0.336i)3-s + (1.51 − 1.30i)4-s + (−2.06 − 1.53i)5-s + (1.41 − 0.0151i)6-s + (−4.35 − 0.428i)7-s + (−1.37 + 2.47i)8-s + (0.773 + 0.634i)9-s + (3.49 + 1.01i)10-s + (−0.812 − 0.897i)11-s + (−1.86 + 0.713i)12-s + (0.927 − 6.25i)13-s + (5.98 − 1.56i)14-s + (1.42 + 2.13i)15-s + (0.611 − 3.95i)16-s + (−2.05 + 3.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.937 + 0.346i)2-s + (−0.543 − 0.194i)3-s + (0.759 − 0.650i)4-s + (−0.924 − 0.685i)5-s + (0.577 − 0.00617i)6-s + (−1.64 − 0.162i)7-s + (−0.486 + 0.873i)8-s + (0.257 + 0.211i)9-s + (1.10 + 0.322i)10-s + (−0.245 − 0.270i)11-s + (−0.539 + 0.206i)12-s + (0.257 − 1.73i)13-s + (1.59 − 0.418i)14-s + (0.369 + 0.552i)15-s + (0.152 − 0.988i)16-s + (−0.497 + 0.744i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.118960 + 0.103858i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.118960 + 0.103858i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.32 - 0.490i)T \) |
| 3 | \( 1 + (0.941 + 0.336i)T \) |
| good | 5 | \( 1 + (2.06 + 1.53i)T + (1.45 + 4.78i)T^{2} \) |
| 7 | \( 1 + (4.35 + 0.428i)T + (6.86 + 1.36i)T^{2} \) |
| 11 | \( 1 + (0.812 + 0.897i)T + (-1.07 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.927 + 6.25i)T + (-12.4 - 3.77i)T^{2} \) |
| 17 | \( 1 + (2.05 - 3.07i)T + (-6.50 - 15.7i)T^{2} \) |
| 19 | \( 1 + (0.950 - 1.58i)T + (-8.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (3.11 - 1.66i)T + (12.7 - 19.1i)T^{2} \) |
| 29 | \( 1 + (-0.975 + 0.0479i)T + (28.8 - 2.84i)T^{2} \) |
| 31 | \( 1 + (-0.673 + 1.62i)T + (-21.9 - 21.9i)T^{2} \) |
| 37 | \( 1 + (-5.39 - 1.35i)T + (32.6 + 17.4i)T^{2} \) |
| 41 | \( 1 + (0.483 - 1.59i)T + (-34.0 - 22.7i)T^{2} \) |
| 43 | \( 1 + (-1.05 - 2.95i)T + (-33.2 + 27.2i)T^{2} \) |
| 47 | \( 1 + (-2.81 + 0.559i)T + (43.4 - 17.9i)T^{2} \) |
| 53 | \( 1 + (0.666 + 0.0327i)T + (52.7 + 5.19i)T^{2} \) |
| 59 | \( 1 + (-1.51 - 10.1i)T + (-56.4 + 17.1i)T^{2} \) |
| 61 | \( 1 + (-7.44 + 3.52i)T + (38.6 - 47.1i)T^{2} \) |
| 67 | \( 1 + (-5.45 - 11.5i)T + (-42.5 + 51.7i)T^{2} \) |
| 71 | \( 1 + (-4.23 - 5.16i)T + (-13.8 + 69.6i)T^{2} \) |
| 73 | \( 1 + (16.4 - 1.61i)T + (71.5 - 14.2i)T^{2} \) |
| 79 | \( 1 + (-1.75 + 8.80i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (17.4 - 4.36i)T + (73.1 - 39.1i)T^{2} \) |
| 89 | \( 1 + (-0.547 - 0.292i)T + (49.4 + 74.0i)T^{2} \) |
| 97 | \( 1 + (-8.05 - 3.33i)T + (68.5 + 68.5i)T^{2} \) |
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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
−10.29344320047989642265654196504, −9.822695443276636740626803733800, −8.590911852090569402037577077978, −8.054597213143352352043120970168, −7.17401015119865750199283386116, −6.13146388007434507416810064432, −5.63905782873934183984613237102, −4.07916396469710435923030403888, −2.86525146132445924574015135880, −0.826990252878568960276519133464,
0.16497137792268088876303160114, 2.37031132896966964808647849389, 3.46610872474010648322779492471, 4.32223529409266290651678790021, 6.18783934526426927404414797832, 6.81712360666206236975525015106, 7.32225993942107070853662724079, 8.667394735525319179071300858342, 9.442620000167783099949304881951, 10.03635828424571452142841551750
