| L(s) = 1 | + (3.17 + 1.83i)5-s + (0.191 + 0.332i)7-s + (1.73 − 1.00i)11-s + (0.397 + 0.229i)13-s + 4.08·17-s − 4.72i·19-s + (−2.97 + 5.15i)23-s + (4.21 + 7.29i)25-s + (−2.03 + 1.17i)29-s + (−0.592 + 1.02i)31-s + 1.40i·35-s − 5.74i·37-s + (−4.75 + 8.23i)41-s + (1.03 − 0.598i)43-s + (3.27 + 5.67i)47-s + ⋯ |
| L(s) = 1 | + (1.41 + 0.819i)5-s + (0.0725 + 0.125i)7-s + (0.524 − 0.302i)11-s + (0.110 + 0.0636i)13-s + 0.990·17-s − 1.08i·19-s + (−0.620 + 1.07i)23-s + (0.842 + 1.45i)25-s + (−0.378 + 0.218i)29-s + (−0.106 + 0.184i)31-s + 0.237i·35-s − 0.944i·37-s + (−0.742 + 1.28i)41-s + (0.158 − 0.0912i)43-s + (0.477 + 0.827i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.02493 + 0.559976i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.02493 + 0.559976i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-3.17 - 1.83i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.191 - 0.332i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.73 + 1.00i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.397 - 0.229i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.08T + 17T^{2} \) |
| 19 | \( 1 + 4.72iT - 19T^{2} \) |
| 23 | \( 1 + (2.97 - 5.15i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.03 - 1.17i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.592 - 1.02i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.74iT - 37T^{2} \) |
| 41 | \( 1 + (4.75 - 8.23i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.03 + 0.598i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.27 - 5.67i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 7.63iT - 53T^{2} \) |
| 59 | \( 1 + (0.603 + 0.348i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.23 - 2.44i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.87 - 5.12i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 + 2.68T + 73T^{2} \) |
| 79 | \( 1 + (-5.35 - 9.28i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.49 + 3.16i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 7.56T + 89T^{2} \) |
| 97 | \( 1 + (2.98 + 5.17i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.07709408295906124295999725311, −9.562968321190175346319010100537, −8.754988545841153434615104379785, −7.53428963314013500922748899388, −6.68974871684791738339252999443, −5.89661563677417967337650285323, −5.20439356144234123170945340516, −3.67395393747989467819344905146, −2.64923457529750904721412418502, −1.51408281178610158693530062685,
1.22954638774019337605523570225, 2.22345753553391468993492509276, 3.77090485959622582505227769583, 4.88608949086692696336343944807, 5.76546576500534586279055685270, 6.37505655191809730056881030585, 7.62560872678354194648984883457, 8.560899096466740193213955598175, 9.321781202255216001290696615257, 10.05428443189617049486731932606
