| L(s) = 1 | + (0.5 + 0.363i)2-s + (1.12 + 0.502i)3-s + (−0.5 − 1.53i)4-s + (−0.5 + 0.866i)5-s + (0.381 + 0.661i)6-s + (−2.83 + 3.14i)7-s + (0.690 − 2.12i)8-s + (−0.985 − 1.09i)9-s + (−0.564 + 0.251i)10-s + (−1.95 − 0.415i)11-s + (0.209 − 1.98i)12-s + (−0.129 − 1.22i)13-s + (−2.56 + 0.544i)14-s + (−1 + 0.726i)15-s + (−1.49 + 1.08i)16-s + (−5.12 + 1.08i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.256i)2-s + (0.651 + 0.290i)3-s + (−0.250 − 0.769i)4-s + (−0.223 + 0.387i)5-s + (0.155 + 0.270i)6-s + (−1.07 + 1.18i)7-s + (0.244 − 0.751i)8-s + (−0.328 − 0.364i)9-s + (−0.178 + 0.0794i)10-s + (−0.589 − 0.125i)11-s + (0.0603 − 0.574i)12-s + (−0.0358 − 0.340i)13-s + (−0.684 + 0.145i)14-s + (−0.258 + 0.187i)15-s + (−0.374 + 0.272i)16-s + (−1.24 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.000743602 + 0.334756i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000743602 + 0.334756i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (-0.5 - 0.363i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.12 - 0.502i)T + (2.00 + 2.22i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.83 - 3.14i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (1.95 + 0.415i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.129 + 1.22i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (5.12 - 1.08i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.233 - 2.22i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (2.38 - 7.33i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (5.85 + 4.25i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.39 + 2.84i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.338 + 3.21i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-5.23 + 3.80i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (1.02 + 1.13i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (2.04 + 0.909i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.81 - 9.79i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-0.461 - 0.0981i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (1.67 - 0.355i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.68 + 1.19i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.527 + 1.62i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.600 - 1.84i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.26111021170137716719032812554, −9.310533564003306302283539267935, −9.168857453122780922389296356695, −8.004903380452685033283792271011, −6.88926599773031379321226716185, −5.87697502562249231931092221967, −5.59002823063009185678774977503, −4.08164493817898073026091465308, −3.24925498478259712805312822736, −2.21704725252552378320276771094,
0.11607423535526923119251637674, 2.33551682088031588461351314441, 3.10166109113870133955242627124, 4.20789034638780961464360002948, 4.75628134886213653314342625747, 6.35403752075445097354471516333, 7.28617286394662353249338386328, 7.87490988847403258260340992007, 8.792890192688992885925462769972, 9.351707584676160220520123063617
