L(s) = 1 | + (−1.11 + 1.53i)2-s + (0.0877 + 1.72i)3-s + (−0.499 − 1.53i)4-s + (−2.76 − 1.79i)6-s + (0.726 + 2.23i)7-s + (−0.690 − 0.224i)8-s + (−2.98 + 0.303i)9-s + (−3.07 + 1.23i)11-s + (2.61 − 0.999i)12-s + (−4.25 − 1.38i)14-s + (3.73 − 2.71i)16-s + (3.61 + 4.97i)17-s + (2.86 − 4.93i)18-s + (−4.30 − 1.40i)19-s + (−3.80 + 1.45i)21-s + (1.53 − 6.11i)22-s + ⋯ |
L(s) = 1 | + (−0.790 + 1.08i)2-s + (0.0506 + 0.998i)3-s + (−0.249 − 0.769i)4-s + (−1.12 − 0.734i)6-s + (0.274 + 0.845i)7-s + (−0.244 − 0.0793i)8-s + (−0.994 + 0.101i)9-s + (−0.927 + 0.372i)11-s + (0.755 − 0.288i)12-s + (−1.13 − 0.369i)14-s + (0.934 − 0.678i)16-s + (0.877 + 1.20i)17-s + (0.676 − 1.16i)18-s + (−0.988 − 0.321i)19-s + (−0.830 + 0.317i)21-s + (0.328 − 1.30i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.324115 - 0.261352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.324115 - 0.261352i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0877 - 1.72i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (3.07 - 1.23i)T \) |
good | 2 | \( 1 + (1.11 - 1.53i)T + (-0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.726 - 2.23i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.61 - 4.97i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (4.30 + 1.40i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 + (2.21 + 6.80i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.04 - 2.93i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (4.47 - 1.45i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.224 - 0.690i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 7.33T + 43T^{2} \) |
| 47 | \( 1 + (-1.57 + 4.84i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (6.16 + 4.47i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-5.20 + 1.69i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-4.89 - 6.74i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 12.3iT - 67T^{2} \) |
| 71 | \( 1 + (6.88 + 9.47i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.224 + 0.690i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-1.54 + 2.12i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-9.47 - 13.0i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 0.854iT - 89T^{2} \) |
| 97 | \( 1 + (-9.99 + 13.7i)T + (-29.9 - 92.2i)T^{2} \) |
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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
−10.28982799343064497595579416350, −10.06929717564646471125799743144, −8.937961317187334956733060359905, −8.358488812997659240360036307301, −7.84173218053102934136805501811, −6.50917900134598619203405577256, −5.72069495528597724484598375883, −4.99304501828399205445259690173, −3.68402649236685877814467378950, −2.38033932452367962127719950642,
0.26421825354052929161705867493, 1.42805961569699641791150076930, 2.56030636323506146672105055590, 3.48945834683231543755542359130, 5.11672680161962732806930899845, 6.16641667854170600941040564118, 7.30665771079849533806419083766, 7.985901670663326224546550925416, 8.705777650416723318472552601284, 9.716431579009749035064873054307