L(s) = 1 | − 2-s + (1.21 − 1.23i)3-s + 4-s + (−1.57 + 0.908i)5-s + (−1.21 + 1.23i)6-s + (−2.47 − 0.947i)7-s − 8-s + (−0.0353 − 2.99i)9-s + (1.57 − 0.908i)10-s + (1.02 + 1.77i)11-s + (1.21 − 1.23i)12-s + (−3.57 − 0.463i)13-s + (2.47 + 0.947i)14-s + (−0.796 + 3.04i)15-s + 16-s − 5.68·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.702 − 0.711i)3-s + 0.5·4-s + (−0.703 + 0.406i)5-s + (−0.497 + 0.502i)6-s + (−0.933 − 0.358i)7-s − 0.353·8-s + (−0.0117 − 0.999i)9-s + (0.497 − 0.287i)10-s + (0.309 + 0.535i)11-s + (0.351 − 0.355i)12-s + (−0.991 − 0.128i)13-s + (0.660 + 0.253i)14-s + (−0.205 + 0.786i)15-s + 0.250·16-s − 1.37·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0103116 + 0.233747i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0103116 + 0.233747i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.21 + 1.23i)T \) |
| 7 | \( 1 + (2.47 + 0.947i)T \) |
| 13 | \( 1 + (3.57 + 0.463i)T \) |
good | 5 | \( 1 + (1.57 - 0.908i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.02 - 1.77i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + (0.796 - 1.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 8.73iT - 23T^{2} \) |
| 29 | \( 1 + (0.724 + 0.418i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.97 - 6.88i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.21iT - 37T^{2} \) |
| 41 | \( 1 + (0.397 + 0.229i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.836 - 1.44i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.94 + 1.12i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.497 + 0.286i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.19iT - 59T^{2} \) |
| 61 | \( 1 + (5.97 + 3.44i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.46 - 2.57i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.24 + 7.34i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.11 + 8.86i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.68 + 6.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 15.0iT - 83T^{2} \) |
| 89 | \( 1 + 3.86iT - 89T^{2} \) |
| 97 | \( 1 + (1.72 + 2.98i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.24259560613899373098634897573, −9.298752010650306570330562241839, −8.631315568246097335149303516332, −7.50746191829973409822572359082, −7.02089776102078591559919591636, −6.31134258976212968605269413496, −4.33860644644166896532665189849, −3.18268876784789527685414788490, −2.12867149833423758270933282953, −0.14253174107031891466824402737,
2.28691559884651409369582360593, 3.44121177141764925613397033315, 4.44435602782295764095082941302, 5.76291354248580493132247750053, 7.04739379527837260151088789806, 7.88721163528356953446657044868, 8.871886775437934056967210666487, 9.319727327121057896112665188508, 10.07624512377497149999251345695, 11.19568772127196106213665919955