L(s) = 1 | + (−1.61 − 2.21i)2-s + (1.24 − 1.24i)3-s + (−1.70 + 5.23i)4-s + (−1.15 − 0.375i)5-s + (−4.75 − 0.753i)6-s + (1.19 − 0.189i)7-s + (9.13 − 2.96i)8-s − 0.0941i·9-s + (1.02 + 3.16i)10-s + (0.836 + 0.426i)11-s + (4.39 + 8.62i)12-s + (0.289 − 1.82i)13-s + (−2.34 − 2.34i)14-s + (−1.90 + 0.969i)15-s + (−12.3 − 8.98i)16-s + (−1.02 + 2.01i)17-s + ⋯ |
L(s) = 1 | + (−1.13 − 1.56i)2-s + (0.718 − 0.718i)3-s + (−0.850 + 2.61i)4-s + (−0.516 − 0.167i)5-s + (−1.94 − 0.307i)6-s + (0.451 − 0.0715i)7-s + (3.22 − 1.04i)8-s − 0.0313i·9-s + (0.324 + 0.999i)10-s + (0.252 + 0.128i)11-s + (1.26 + 2.49i)12-s + (0.0802 − 0.506i)13-s + (−0.626 − 0.626i)14-s + (−0.491 + 0.250i)15-s + (−3.09 − 2.24i)16-s + (−0.248 + 0.488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.272055 - 0.464598i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.272055 - 0.464598i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 + (-4.89 + 4.13i)T \) |
good | 2 | \( 1 + (1.61 + 2.21i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.24 + 1.24i)T - 3iT^{2} \) |
| 5 | \( 1 + (1.15 + 0.375i)T + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (-1.19 + 0.189i)T + (6.65 - 2.16i)T^{2} \) |
| 11 | \( 1 + (-0.836 - 0.426i)T + (6.46 + 8.89i)T^{2} \) |
| 13 | \( 1 + (-0.289 + 1.82i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (1.02 - 2.01i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.815 - 5.15i)T + (-18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (5.31 - 3.86i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.689 + 1.35i)T + (-17.0 + 23.4i)T^{2} \) |
| 31 | \( 1 + (0.515 + 1.58i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.97 + 6.06i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-0.714 - 0.982i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (5.36 + 0.849i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (4.56 + 8.95i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (6.23 - 4.52i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.982 + 1.35i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-13.1 + 6.68i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-2.77 - 1.41i)T + (41.7 + 57.4i)T^{2} \) |
| 73 | \( 1 - 0.596iT - 73T^{2} \) |
| 79 | \( 1 + (5.89 - 5.89i)T - 79iT^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + (-6.08 + 0.964i)T + (84.6 - 27.5i)T^{2} \) |
| 97 | \( 1 + (15.0 - 7.65i)T + (57.0 - 78.4i)T^{2} \) |
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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
−16.16468170434835274478985687004, −14.14346566609883630783092065711, −12.94870414072639491323500077310, −12.06612983880110747250697065609, −10.93975444980216376041209332288, −9.641219432062798780502879386462, −8.173206698376730060135772989523, −7.79298909044953836633838839117, −3.79708842512108554983900580860, −1.91115962881095603746001628930,
4.60918941050621403651065942239, 6.52932312526660619031835495947, 7.894931926478431279323471508741, 8.911420429040030354950456136904, 9.776692837590448855972138989618, 11.24324846493893354648720106063, 13.91030712045174478060010767639, 14.70489309577474725263301015235, 15.55858428697906511757327271478, 16.24497332730710166356632846383