| L(s) = 1 | + (0.280 − 0.649i)2-s + (1.02 + 1.09i)4-s + (−0.199 − 3.42i)5-s + (2.63 + 0.623i)7-s + (2.32 − 0.846i)8-s + (−2.28 − 0.831i)10-s + (−1.23 − 0.811i)11-s + (5.51 + 0.644i)13-s + (1.14 − 1.53i)14-s + (−0.0725 + 1.24i)16-s + (−3.86 + 3.24i)17-s + (−3.94 − 3.30i)19-s + (3.53 − 3.74i)20-s + (−0.873 + 0.574i)22-s + (1.68 − 0.399i)23-s + ⋯ |
| L(s) = 1 | + (0.198 − 0.459i)2-s + (0.514 + 0.545i)4-s + (−0.0893 − 1.53i)5-s + (0.994 + 0.235i)7-s + (0.822 − 0.299i)8-s + (−0.722 − 0.262i)10-s + (−0.372 − 0.244i)11-s + (1.52 + 0.178i)13-s + (0.305 − 0.410i)14-s + (−0.0181 + 0.311i)16-s + (−0.937 + 0.786i)17-s + (−0.904 − 0.758i)19-s + (0.790 − 0.837i)20-s + (−0.186 + 0.122i)22-s + (0.351 − 0.0833i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88892 - 1.09809i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88892 - 1.09809i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.280 + 0.649i)T + (-1.37 - 1.45i)T^{2} \) |
| 5 | \( 1 + (0.199 + 3.42i)T + (-4.96 + 0.580i)T^{2} \) |
| 7 | \( 1 + (-2.63 - 0.623i)T + (6.25 + 3.14i)T^{2} \) |
| 11 | \( 1 + (1.23 + 0.811i)T + (4.35 + 10.1i)T^{2} \) |
| 13 | \( 1 + (-5.51 - 0.644i)T + (12.6 + 2.99i)T^{2} \) |
| 17 | \( 1 + (3.86 - 3.24i)T + (2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (3.94 + 3.30i)T + (3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.68 + 0.399i)T + (20.5 - 10.3i)T^{2} \) |
| 29 | \( 1 + (-1.28 - 1.72i)T + (-8.31 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-1.09 + 3.64i)T + (-25.9 - 17.0i)T^{2} \) |
| 37 | \( 1 + (0.891 + 5.05i)T + (-34.7 + 12.6i)T^{2} \) |
| 41 | \( 1 + (1.78 + 4.14i)T + (-28.1 + 29.8i)T^{2} \) |
| 43 | \( 1 + (-0.736 + 0.370i)T + (25.6 - 34.4i)T^{2} \) |
| 47 | \( 1 + (-1.92 - 6.43i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-2.58 - 4.48i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.95 + 1.94i)T + (23.3 - 54.1i)T^{2} \) |
| 61 | \( 1 + (-3.71 + 3.94i)T + (-3.54 - 60.8i)T^{2} \) |
| 67 | \( 1 + (3.91 - 5.25i)T + (-19.2 - 64.1i)T^{2} \) |
| 71 | \( 1 + (-3.30 - 1.20i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-0.668 + 0.243i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (6.21 - 14.4i)T + (-54.2 - 57.4i)T^{2} \) |
| 83 | \( 1 + (-2.35 + 5.45i)T + (-56.9 - 60.3i)T^{2} \) |
| 89 | \( 1 + (3.78 - 1.37i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.700 + 12.0i)T + (-96.3 - 11.2i)T^{2} \) |
| 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
−10.71328363857410887383530447051, −9.064726103870312949391946248766, −8.495267812100355547323028019189, −8.040253785984663832745601488566, −6.71750633432003784388768777195, −5.60408107636722846330581024230, −4.52476572030650732210229395341, −3.93275696336419853981170890809, −2.28458947642796521904522559839, −1.25111789012192581166268510287,
1.67600786416095111827899047249, 2.83355231997359164681245153646, 4.17420911500723743365889250285, 5.29330371909516154034940934764, 6.38117183918477818027597256402, 6.82099083518734536630062833103, 7.76372589539509417865559032815, 8.540675887450069334096357953306, 10.06255054911948469649607743901, 10.70245496951857951815948583781
