L(s) = 1 | + (0.721 + 1.57i)3-s + (0.976 − 0.172i)5-s + (3.12 − 3.72i)7-s + (−1.95 + 2.27i)9-s + (0.961 − 5.45i)11-s + (2.65 + 0.966i)13-s + (0.976 + 1.41i)15-s + (−0.517 + 0.298i)17-s + (0.141 + 0.0815i)19-s + (8.12 + 2.23i)21-s + (−5.35 + 4.49i)23-s + (−3.77 + 1.37i)25-s + (−4.99 − 1.44i)27-s + (0.610 + 1.67i)29-s + (3.26 + 3.88i)31-s + ⋯ |
L(s) = 1 | + (0.416 + 0.909i)3-s + (0.436 − 0.0770i)5-s + (1.18 − 1.40i)7-s + (−0.652 + 0.757i)9-s + (0.289 − 1.64i)11-s + (0.736 + 0.268i)13-s + (0.252 + 0.364i)15-s + (−0.125 + 0.0724i)17-s + (0.0324 + 0.0187i)19-s + (1.77 + 0.487i)21-s + (−1.11 + 0.937i)23-s + (−0.754 + 0.274i)25-s + (−0.960 − 0.277i)27-s + (0.113 + 0.311i)29-s + (0.586 + 0.698i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.86415 + 0.171973i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.86415 + 0.171973i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.721 - 1.57i)T \) |
good | 5 | \( 1 + (-0.976 + 0.172i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-3.12 + 3.72i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.961 + 5.45i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.65 - 0.966i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.517 - 0.298i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.141 - 0.0815i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.35 - 4.49i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.610 - 1.67i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.26 - 3.88i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-0.726 - 1.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.84 - 10.5i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-6.11 - 1.07i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (6.37 + 5.35i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 8.40iT - 53T^{2} \) |
| 59 | \( 1 + (1.35 + 7.69i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (3.16 + 2.65i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.02 + 5.55i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.86 - 3.23i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.05 + 5.30i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.05 - 11.1i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (9.04 - 3.29i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (1.45 + 0.837i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.34 - 13.2i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.12442921648589131663296893140, −10.35102679049337637968992830030, −9.442799665090024097006607799749, −8.345762076726637386205558878793, −7.902514748398543571951692424258, −6.35243543237569221058395203705, −5.28031408864911452758015093198, −4.17769745586660587033797245193, −3.39181304908104641146758130826, −1.45996969144846080097959087859,
1.82184862454062095840511335339, 2.39873669568142812455807492425, 4.30281703217460852491609861260, 5.59874034831362440546328489708, 6.38306223631024232014163954233, 7.59770670970057077494409721570, 8.343628192878527428073086058484, 9.117691819003292366331952004152, 10.10232219127556483651834343517, 11.45045600234784666237880336030