L(s) = 1 | + 1.90·2-s + 1.63·4-s + (−0.736 + 1.27i)5-s + (1.58 + 2.11i)7-s − 0.702·8-s + (−1.40 + 2.43i)10-s + (−2.19 + 3.80i)11-s + (2.69 + 2.39i)13-s + (3.01 + 4.03i)14-s − 4.60·16-s + 1.20·17-s + (−1.62 − 2.80i)19-s + (−1.20 + 2.08i)20-s + (−4.18 + 7.25i)22-s + 4.43·23-s + ⋯ |
L(s) = 1 | + 1.34·2-s + 0.815·4-s + (−0.329 + 0.570i)5-s + (0.598 + 0.801i)7-s − 0.248·8-s + (−0.443 + 0.768i)10-s + (−0.662 + 1.14i)11-s + (0.748 + 0.663i)13-s + (0.806 + 1.07i)14-s − 1.15·16-s + 0.291·17-s + (−0.371 − 0.644i)19-s + (−0.268 + 0.465i)20-s + (−0.892 + 1.54i)22-s + 0.925·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.272 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.19946 + 1.66309i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19946 + 1.66309i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.58 - 2.11i)T \) |
| 13 | \( 1 + (-2.69 - 2.39i)T \) |
good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 5 | \( 1 + (0.736 - 1.27i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.19 - 3.80i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 1.20T + 17T^{2} \) |
| 19 | \( 1 + (1.62 + 2.80i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + (-0.0837 - 0.145i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.62 + 4.54i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.05T + 37T^{2} \) |
| 41 | \( 1 + (-2.58 - 4.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0113 - 0.0197i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.84 + 10.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0708 + 0.122i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 5.34T + 59T^{2} \) |
| 61 | \( 1 + (5.77 + 9.99i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.06 - 3.58i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.98 - 8.63i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.62 + 13.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.387 - 0.670i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 6.55T + 89T^{2} \) |
| 97 | \( 1 + (1.74 - 3.02i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.84012964599943772882880321208, −9.509497203114606019904346260291, −8.745775969729148364226302253345, −7.58245378107591631691737921783, −6.76965509810713636462014219386, −5.80812753404416147055138734588, −4.94003119276623843151206227937, −4.22289259858948922876531032197, −3.04723527983199537025488103579, −2.11798499969535451287421936931,
0.924449515598442546240690064625, 2.88095485010120275299664620678, 3.79269554824492492579312717069, 4.60879857081086962804100873763, 5.48128460182045803289389582368, 6.17404812855688997167218890997, 7.45626659590684916322453503843, 8.291168655778947441630077306609, 8.991671216562117790953333733816, 10.52747159721385616894192105007