| L(s) = 1 | + (−0.127 − 1.40i)2-s + (0.881 + 0.471i)3-s + (−1.96 + 0.359i)4-s + (−1.40 + 1.71i)5-s + (0.551 − 1.30i)6-s + (0.0127 + 0.00855i)7-s + (0.758 + 2.72i)8-s + (0.555 + 0.831i)9-s + (2.59 + 1.76i)10-s + (3.40 + 1.03i)11-s + (−1.90 − 0.610i)12-s + (−2.41 + 1.98i)13-s + (0.0104 − 0.0191i)14-s + (−2.04 + 0.848i)15-s + (3.74 − 1.41i)16-s + (4.96 + 2.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.0903 − 0.995i)2-s + (0.509 + 0.272i)3-s + (−0.983 + 0.179i)4-s + (−0.629 + 0.766i)5-s + (0.225 − 0.531i)6-s + (0.00483 + 0.00323i)7-s + (0.268 + 0.963i)8-s + (0.185 + 0.277i)9-s + (0.820 + 0.557i)10-s + (1.02 + 0.311i)11-s + (−0.549 − 0.176i)12-s + (−0.670 + 0.550i)13-s + (0.00278 − 0.00510i)14-s + (−0.529 + 0.219i)15-s + (0.935 − 0.353i)16-s + (1.20 + 0.499i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21632 + 0.177328i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21632 + 0.177328i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.127 + 1.40i)T \) |
| 3 | \( 1 + (-0.881 - 0.471i)T \) |
| good | 5 | \( 1 + (1.40 - 1.71i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-0.0127 - 0.00855i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-3.40 - 1.03i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (2.41 - 1.98i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.96 - 2.05i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.321 - 3.26i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-4.07 - 0.809i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (0.827 + 2.72i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (3.88 - 3.88i)T - 31iT^{2} \) |
| 37 | \( 1 + (2.61 - 0.257i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (-1.20 + 6.07i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (9.65 - 5.16i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-2.67 + 6.45i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.861 - 2.84i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-5.35 - 4.39i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-5.71 + 10.6i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (-2.05 + 3.83i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (1.74 - 2.61i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-6.90 + 4.61i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.304 - 0.734i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (5.51 + 0.543i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (6.60 - 1.31i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-10.0 + 10.0i)T - 97iT^{2} \) |
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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
−11.43802875090498761410333190186, −10.41481872618459933380417712740, −9.736814272604654114646880308413, −8.842114541443085767899189682803, −7.82950927137054673653614749451, −6.88867359809014541015714337304, −5.19895930402404071268901013245, −3.88932087137612894784119966176, −3.31662301241413692925169780592, −1.79266521355358861184362368634,
0.893711259457127491507841946860, 3.32816058742216865636440875617, 4.51603467112180805735002529884, 5.47949169287321000975468563421, 6.80585766163428859855840519580, 7.58173487341169722863446055702, 8.440491802546204230945941561425, 9.137438814524293562191020623186, 9.966708246337619024361651029886, 11.47127349786829451642956852774
