| L(s) = 1 | + (−0.844 − 0.192i)2-s + (4.86 − 1.10i)3-s + (−2.92 − 1.40i)4-s + (0.831 + 0.663i)5-s − 4.32·6-s − 0.302i·7-s + (4.91 + 3.91i)8-s + (14.3 − 6.88i)9-s + (−0.574 − 0.720i)10-s + (−4.95 + 2.38i)11-s + (−15.8 − 3.60i)12-s + (−12.6 + 15.8i)13-s + (−0.0582 + 0.255i)14-s + (4.77 + 2.30i)15-s + (4.71 + 5.90i)16-s + (−13.9 − 17.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.422 − 0.0963i)2-s + (1.62 − 0.369i)3-s + (−0.731 − 0.352i)4-s + (0.166 + 0.132i)5-s − 0.720·6-s − 0.0431i·7-s + (0.613 + 0.489i)8-s + (1.58 − 0.765i)9-s + (−0.0574 − 0.0720i)10-s + (−0.450 + 0.217i)11-s + (−1.31 − 0.300i)12-s + (−0.972 + 1.21i)13-s + (−0.00416 + 0.0182i)14-s + (0.318 + 0.153i)15-s + (0.294 + 0.369i)16-s + (−0.819 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.16111 - 0.287192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16111 - 0.287192i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (16.1 + 39.8i)T \) |
| good | 2 | \( 1 + (0.844 + 0.192i)T + (3.60 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-4.86 + 1.10i)T + (8.10 - 3.90i)T^{2} \) |
| 5 | \( 1 + (-0.831 - 0.663i)T + (5.56 + 24.3i)T^{2} \) |
| 7 | \( 1 + 0.302iT - 49T^{2} \) |
| 11 | \( 1 + (4.95 - 2.38i)T + (75.4 - 94.6i)T^{2} \) |
| 13 | \( 1 + (12.6 - 15.8i)T + (-37.6 - 164. i)T^{2} \) |
| 17 | \( 1 + (13.9 + 17.4i)T + (-64.3 + 281. i)T^{2} \) |
| 19 | \( 1 + (11.5 - 23.9i)T + (-225. - 282. i)T^{2} \) |
| 23 | \( 1 + (-25.4 + 12.2i)T + (329. - 413. i)T^{2} \) |
| 29 | \( 1 + (-32.2 - 7.37i)T + (757. + 364. i)T^{2} \) |
| 31 | \( 1 + (-1.10 + 4.83i)T + (-865. - 416. i)T^{2} \) |
| 37 | \( 1 - 24.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-12.5 + 55.0i)T + (-1.51e3 - 729. i)T^{2} \) |
| 47 | \( 1 + (11.2 + 5.40i)T + (1.37e3 + 1.72e3i)T^{2} \) |
| 53 | \( 1 + (-39.6 - 49.7i)T + (-625. + 2.73e3i)T^{2} \) |
| 59 | \( 1 + (51.6 + 64.7i)T + (-774. + 3.39e3i)T^{2} \) |
| 61 | \( 1 + (50.9 - 11.6i)T + (3.35e3 - 1.61e3i)T^{2} \) |
| 67 | \( 1 + (35.8 + 17.2i)T + (2.79e3 + 3.50e3i)T^{2} \) |
| 71 | \( 1 + (9.29 - 19.2i)T + (-3.14e3 - 3.94e3i)T^{2} \) |
| 73 | \( 1 + (-72.6 - 57.9i)T + (1.18e3 + 5.19e3i)T^{2} \) |
| 79 | \( 1 + 79.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-21.7 - 95.4i)T + (-6.20e3 + 2.98e3i)T^{2} \) |
| 89 | \( 1 + (29.9 - 6.84i)T + (7.13e3 - 3.43e3i)T^{2} \) |
| 97 | \( 1 + (-32.3 + 15.5i)T + (5.86e3 - 7.35e3i)T^{2} \) |
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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
−15.27787695002477216894393041674, −14.17926555172719118827435209448, −13.79302971236987927791502436629, −12.42327864558352774373526929419, −10.30976172996571988319228270325, −9.272665476738427859153628518487, −8.428370797848724576778395972002, −7.08450035071654198669819914338, −4.47990582925586600571716430136, −2.27982692243554771628956419717,
2.97357998256302860444905113530, 4.67368161112808650548262841087, 7.53455915435116830161913705763, 8.525497642715799693365534939417, 9.349974914360681681332607297781, 10.48708305519881136701262311188, 12.98601687393868135614595751285, 13.35913193712836832924471821389, 14.81893236427436257648592376424, 15.49843801306352605206431062092
