| L(s) = 1 | + (−0.313 + 2.18i)2-s + (−1.04 − 2.28i)3-s + (−2.74 − 0.804i)4-s + (−0.809 − 0.934i)5-s + (5.31 − 1.56i)6-s + (1.99 + 1.28i)7-s + (0.783 − 1.71i)8-s + (−2.17 + 2.50i)9-s + (2.29 − 1.47i)10-s + (−0.272 − 1.89i)11-s + (1.02 + 7.10i)12-s + (0.165 − 0.106i)13-s + (−3.42 + 3.95i)14-s + (−1.29 + 2.82i)15-s + (−1.30 − 0.840i)16-s + (−1.49 + 0.439i)17-s + ⋯ |
| L(s) = 1 | + (−0.221 + 1.54i)2-s + (−0.602 − 1.31i)3-s + (−1.37 − 0.402i)4-s + (−0.362 − 0.417i)5-s + (2.16 − 0.637i)6-s + (0.754 + 0.484i)7-s + (0.277 − 0.606i)8-s + (−0.724 + 0.835i)9-s + (0.724 − 0.465i)10-s + (−0.0820 − 0.570i)11-s + (0.294 + 2.05i)12-s + (0.0458 − 0.0294i)13-s + (−0.915 + 1.05i)14-s + (−0.333 + 0.729i)15-s + (−0.327 − 0.210i)16-s + (−0.363 + 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.282666 - 0.286567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.282666 - 0.286567i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (0.313 - 2.18i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (1.04 + 2.28i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (0.809 + 0.934i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.99 - 1.28i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.272 + 1.89i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.165 + 0.106i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.49 - 0.439i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (7.66 + 2.24i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (4.77 - 1.40i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.740 + 1.62i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-2.54 + 2.93i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.279 + 0.322i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.84 + 4.05i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + (8.26 + 5.30i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (6.07 - 3.90i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (3.08 - 6.75i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.03 + 7.19i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.103 + 0.722i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.18 - 1.81i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-4.77 + 3.06i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (8.44 - 9.74i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-5.77 - 12.6i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-2.83 - 3.26i)T + (-13.8 + 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.87990892131619578468809621490, −9.138596769304553075092618686603, −8.332806334120966291660137590259, −7.933089532437944457766797354071, −6.88724027006671575654605310962, −6.25009870872336358486214986666, −5.44963360823369637965391708045, −4.45218052770469246444425629502, −2.08185121065114011654077729851, −0.26548972150234118382937908098,
1.84181793046640651356725609937, 3.35140824077110391928742623866, 4.28351810738368491045015265474, 4.77820176569572945425736352871, 6.33190514143684341975296311407, 7.74363655909304753318756850244, 8.898145689130438029817743900028, 9.776610350042538888525956908178, 10.41712751636922791666227677399, 11.15343797347279074112663290005
