| L(s) = 1 | + (−2.16 − 0.459i)2-s + (0.943 + 1.63i)3-s + (2.63 + 1.17i)4-s + (0.323 + 3.07i)5-s + (−1.28 − 3.96i)6-s + (−1.43 + 2.22i)7-s + (−1.58 − 1.15i)8-s + (−0.279 + 0.484i)9-s + (0.715 − 6.80i)10-s + (−0.475 + 4.52i)11-s + (0.569 + 5.42i)12-s + (−0.814 − 2.50i)13-s + (4.11 − 4.15i)14-s + (−4.72 + 3.43i)15-s + (−0.958 − 1.06i)16-s + (0.558 − 5.30i)17-s + ⋯ |
| L(s) = 1 | + (−1.52 − 0.325i)2-s + (0.544 + 0.943i)3-s + (1.31 + 0.587i)4-s + (0.144 + 1.37i)5-s + (−0.526 − 1.61i)6-s + (−0.540 + 0.841i)7-s + (−0.562 − 0.408i)8-s + (−0.0933 + 0.161i)9-s + (0.226 − 2.15i)10-s + (−0.143 + 1.36i)11-s + (0.164 + 1.56i)12-s + (−0.225 − 0.695i)13-s + (1.10 − 1.11i)14-s + (−1.21 + 0.886i)15-s + (−0.239 − 0.266i)16-s + (0.135 − 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.240173 + 0.570407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.240173 + 0.570407i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.43 - 2.22i)T \) |
| 41 | \( 1 + (3.02 - 5.64i)T \) |
| good | 2 | \( 1 + (2.16 + 0.459i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.943 - 1.63i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.323 - 3.07i)T + (-4.89 + 1.03i)T^{2} \) |
| 11 | \( 1 + (0.475 - 4.52i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.814 + 2.50i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.558 + 5.30i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (3.08 + 3.42i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-4.77 - 1.01i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (5.24 - 3.81i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.888 - 8.45i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.502 + 4.78i)T + (-36.1 + 7.69i)T^{2} \) |
| 43 | \( 1 + (0.546 + 1.68i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-8.64 - 1.83i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-4.85 - 2.16i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (2.41 - 2.68i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-5.88 - 6.54i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (1.58 + 0.704i)T + (44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (2.74 + 1.99i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.94 - 3.37i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.53 + 14.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 5.63T + 83T^{2} \) |
| 89 | \( 1 + (2.64 + 2.94i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (1.94 - 1.41i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.73123764865961947989672350315, −10.60343282384453254795876993173, −10.24785985535371537247798470369, −9.294026660797184928458740409781, −8.949286303331470561050623785380, −7.35953557297805317696718222811, −6.86990873617220276834318272859, −5.01485114108682910077576554505, −3.13157477115636970254003207677, −2.44942177565737672509058359586,
0.71709409327777883428542013075, 1.85983498437900514866401159939, 4.04020882670220987820218085598, 5.92266907654112349416264423831, 6.93754948561632786519708377364, 7.992031363743357296837455740187, 8.442974635064051525198833987298, 9.230997907268220232462181853195, 10.23104855326462141282053477764, 11.18793638758125758996855139784
