| L(s) = 1 | + (−0.316 − 1.37i)2-s + (−1.18 − 2.59i)3-s + (−1.79 + 0.873i)4-s + (1.01 + 1.17i)5-s + (−3.20 + 2.45i)6-s + (−4.25 − 2.73i)7-s + (1.77 + 2.20i)8-s + (−3.37 + 3.89i)9-s + (1.29 − 1.77i)10-s + (1.77 − 0.254i)11-s + (4.40 + 3.63i)12-s + (0.511 + 0.796i)13-s + (−2.41 + 6.73i)14-s + (1.84 − 4.03i)15-s + (2.47 − 3.14i)16-s + (−1.67 − 5.70i)17-s + ⋯ |
| L(s) = 1 | + (−0.224 − 0.974i)2-s + (−0.685 − 1.50i)3-s + (−0.899 + 0.436i)4-s + (0.454 + 0.524i)5-s + (−1.30 + 1.00i)6-s + (−1.60 − 1.03i)7-s + (0.627 + 0.778i)8-s + (−1.12 + 1.29i)9-s + (0.409 − 0.560i)10-s + (0.534 − 0.0768i)11-s + (1.27 + 1.05i)12-s + (0.141 + 0.220i)13-s + (−0.646 + 1.79i)14-s + (0.475 − 1.04i)15-s + (0.618 − 0.785i)16-s + (−0.406 − 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.180282 + 0.458978i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.180282 + 0.458978i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.316 + 1.37i)T \) |
| 23 | \( 1 + (3.08 - 3.67i)T \) |
| good | 3 | \( 1 + (1.18 + 2.59i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-1.01 - 1.17i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (4.25 + 2.73i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-1.77 + 0.254i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.511 - 0.796i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.67 + 5.70i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.812 - 2.76i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (1.87 + 6.39i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.15 + 0.982i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-3.98 + 4.59i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.66 + 4.23i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-5.98 + 2.73i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 4.26iT - 47T^{2} \) |
| 53 | \( 1 + (3.03 + 1.94i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.53 + 1.63i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (2.27 - 4.99i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 1.50i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-2.06 - 0.296i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (2.05 + 0.603i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-6.55 + 4.21i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-6.06 - 5.25i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-2.48 + 1.13i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-0.549 + 0.476i)T + (13.8 - 96.0i)T^{2} \) |
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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
−12.04875618619718258704238927984, −11.20639514114950474789725554528, −10.14651137154035513274742333182, −9.334898498558556599505381992246, −7.63757148991208994907619529037, −6.82153927698031654354868993842, −5.90581127319899043511387798585, −3.80539116559312044898949733304, −2.29163414614900226182965843889, −0.51212435766874151947875335464,
3.63992812180613918724219979234, 4.89331537983396940667601482683, 5.93347412114921824132366292199, 6.45847573436712063873303241008, 8.651558385202763232032734394188, 9.280791210231867785665328646294, 9.893030847598030388863922794366, 10.88763338232981397891208699639, 12.48103433638003052385880930119, 13.10766452447821361227706413246
