L(s) = 1 | + (0.0646 − 0.614i)2-s + (0.669 + 0.743i)3-s + (1.58 + 0.336i)4-s + (−2.04 − 0.909i)5-s + (0.5 − 0.363i)6-s + (−0.0510 + 2.64i)7-s + (0.690 − 2.12i)8-s + (0.209 − 1.98i)9-s + (−0.690 + 1.19i)10-s + (−3.28 − 0.468i)11-s + (0.809 + 1.40i)12-s + (−0.690 − 0.502i)13-s + (1.62 + 0.202i)14-s + (−0.690 − 2.12i)15-s + (1.69 + 0.754i)16-s + (0.129 + 1.22i)17-s + ⋯ |
L(s) = 1 | + (0.0456 − 0.434i)2-s + (0.386 + 0.429i)3-s + (0.791 + 0.168i)4-s + (−0.913 − 0.406i)5-s + (0.204 − 0.148i)6-s + (−0.0193 + 0.999i)7-s + (0.244 − 0.751i)8-s + (0.0696 − 0.663i)9-s + (−0.218 + 0.378i)10-s + (−0.989 − 0.141i)11-s + (0.233 + 0.404i)12-s + (−0.191 − 0.139i)13-s + (0.433 + 0.0540i)14-s + (−0.178 − 0.549i)15-s + (0.423 + 0.188i)16-s + (0.0313 + 0.298i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06779 - 0.107465i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06779 - 0.107465i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.0510 - 2.64i)T \) |
| 11 | \( 1 + (3.28 + 0.468i)T \) |
good | 2 | \( 1 + (-0.0646 + 0.614i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (-0.669 - 0.743i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (2.04 + 0.909i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (0.690 + 0.502i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.129 - 1.22i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (7.68 - 1.63i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.11 - 5.40i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.354 + 1.08i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.04 + 2.69i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-3.40 + 3.78i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (1.42 - 4.39i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 + (-8.28 + 1.76i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (4.65 - 2.07i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (3.62 + 0.770i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.34 - 2.82i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (5.30 - 9.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.23 - 3.07i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.05 + 0.861i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (0.0740 - 0.704i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-7.28 + 5.29i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (5.23 + 9.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.30 + 1.67i)T + (29.9 + 92.2i)T^{2} \) |
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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
−14.95516555744008996706546179204, −12.93967645732501353846469808417, −12.24643067071255628882367470666, −11.31157546973174299255257915655, −10.11734476134993122224272346261, −8.745891394345849694780047311016, −7.73398546643589173907339906710, −6.05871841165443151200266022321, −4.13766629233040797137935539461, −2.70417713291401016420576223776,
2.60552208919929448933064655200, 4.65578688271586897474356695260, 6.67685639934637494595092862471, 7.48620161201420600419219484261, 8.259770906611947056981526685979, 10.56165326944677098764514839300, 10.93426192577923284147111555444, 12.46001783540822941930696431084, 13.59776939670073500259098996765, 14.65512497385260678762279790358