| L(s) = 1 | + (1.05 − 1.22i)2-s + (1.88 − 1.20i)3-s + (−0.0879 − 0.611i)4-s + (−1.34 − 2.94i)5-s + (0.514 − 3.58i)6-s + (1.18 − 0.348i)7-s + (1.88 + 1.20i)8-s + (0.830 − 1.81i)9-s + (−5.02 − 1.47i)10-s + (0.500 + 0.577i)11-s + (−0.904 − 1.04i)12-s + (−2.87 − 0.845i)13-s + (0.830 − 1.81i)14-s + (−6.08 − 3.91i)15-s + (4.65 − 1.36i)16-s + (−0.745 + 5.18i)17-s + ⋯ |
| L(s) = 1 | + (0.749 − 0.864i)2-s + (1.08 − 0.697i)3-s + (−0.0439 − 0.305i)4-s + (−0.601 − 1.31i)5-s + (0.210 − 1.46i)6-s + (0.448 − 0.131i)7-s + (0.665 + 0.427i)8-s + (0.276 − 0.606i)9-s + (−1.58 − 0.466i)10-s + (0.150 + 0.174i)11-s + (−0.261 − 0.301i)12-s + (−0.798 − 0.234i)13-s + (0.222 − 0.486i)14-s + (−1.57 − 1.01i)15-s + (1.16 − 0.341i)16-s + (−0.180 + 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.47864 - 2.43276i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.47864 - 2.43276i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.05 + 1.22i)T + (-0.284 - 1.97i)T^{2} \) |
| 3 | \( 1 + (-1.88 + 1.20i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (1.34 + 2.94i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (-1.18 + 0.348i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (-0.500 - 0.577i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.87 + 0.845i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.745 - 5.18i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.284 - 1.97i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.426 + 2.96i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (5.64 + 3.62i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-1.34 + 2.94i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-2.27 - 4.97i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + (-8.12 + 2.38i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-2.37 - 0.696i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-9.20 - 5.91i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-4.73 + 5.46i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (5.08 - 5.86i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (2.20 + 15.3i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (6.66 + 1.95i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (5.49 - 12.0i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (1.28 - 0.826i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.78 - 3.90i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.83478321208586024063874035018, −9.619982129096495696612306899022, −8.505923120347427883001575484111, −8.040921024720272644285505223826, −7.34171960902043661955260799045, −5.52251916577137733029206657915, −4.44843824407139561424552301440, −3.75858566041896696879005946603, −2.39811987723132095052995566772, −1.43665508999697711885941810589,
2.55251207550192598626392230033, 3.51254241738113271999290420492, 4.44408452102152168683858046340, 5.42199074416971689939333079774, 6.92013987900015090251679447826, 7.20020758268929726339027778380, 8.282479049555080997284230308197, 9.333003436868723181716017055836, 10.19361204299943238738225638025, 11.06048606432681709333967168144
