L(s) = 1 | + (−2.01 + 1.16i)2-s + (1.21 + 1.23i)3-s + (1.71 − 2.97i)4-s + (−0.5 − 0.866i)5-s + (−3.89 − 1.07i)6-s + 3.33i·8-s + (−0.0404 + 2.99i)9-s + (2.01 + 1.16i)10-s + (2.42 + 1.39i)11-s + (5.75 − 1.50i)12-s − 3.20i·13-s + (0.459 − 1.66i)15-s + (−0.459 − 0.795i)16-s + (0.440 − 0.763i)17-s + (−3.41 − 6.10i)18-s + (−1.90 + 1.09i)19-s + ⋯ |
L(s) = 1 | + (−1.42 + 0.824i)2-s + (0.702 + 0.711i)3-s + (0.858 − 1.48i)4-s + (−0.223 − 0.387i)5-s + (−1.58 − 0.437i)6-s + 1.18i·8-s + (−0.0134 + 0.999i)9-s + (0.638 + 0.368i)10-s + (0.729 + 0.421i)11-s + (1.66 − 0.433i)12-s − 0.888i·13-s + (0.118 − 0.431i)15-s + (−0.114 − 0.198i)16-s + (0.106 − 0.185i)17-s + (−0.804 − 1.43i)18-s + (−0.436 + 0.251i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.595329 + 0.721969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.595329 + 0.721969i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.21 - 1.23i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (2.01 - 1.16i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.42 - 1.39i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.20iT - 13T^{2} \) |
| 17 | \( 1 + (-0.440 + 0.763i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.90 - 1.09i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.53 + 3.77i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.15iT - 29T^{2} \) |
| 31 | \( 1 + (-7.62 - 4.40i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.203 + 0.352i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.55T + 41T^{2} \) |
| 43 | \( 1 + 0.118T + 43T^{2} \) |
| 47 | \( 1 + (-1.31 - 2.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.46 - 3.73i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.04 - 3.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.7 - 6.17i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.802 + 1.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.25iT - 71T^{2} \) |
| 73 | \( 1 + (0.192 + 0.110i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.56 - 2.71i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.666T + 83T^{2} \) |
| 89 | \( 1 + (-0.437 - 0.757i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.37iT - 97T^{2} \) |
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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
−10.49457450849976913625904308921, −9.349404614013370607764712943858, −8.961492012794243474343661231159, −8.228754882040086332322000918615, −7.46486066144415292241468959700, −6.58884974577771066685320462185, −5.34318047666484641369403329628, −4.30465831815547654574559119322, −2.91256233140021247629891367188, −1.20172590220257949054379174153,
0.873888480151054599464346343226, 2.08587513206889237162603826331, 3.02317775544125398819292326566, 4.11785668307936204752119175204, 6.17967667487642135653647884420, 7.03636297064954881455385766539, 7.82009950691214535720312648664, 8.586811464719764412290528457286, 9.281704528653825219764409836006, 9.860988341767447080150628511655