L(s) = 1 | + (0.169 − 1.40i)2-s + (−0.416 + 0.240i)3-s + (−1.94 − 0.477i)4-s + (1.59 − 2.76i)5-s + (0.266 + 0.625i)6-s + (0.694 + 2.55i)7-s + (−0.999 + 2.64i)8-s + (−1.38 + 2.39i)9-s + (−3.61 − 2.71i)10-s + (0.800 + 1.38i)11-s + (0.923 − 0.268i)12-s − 1.38·13-s + (3.70 − 0.540i)14-s + 1.53i·15-s + (3.54 + 1.85i)16-s + (3.48 − 2.01i)17-s + ⋯ |
L(s) = 1 | + (0.120 − 0.992i)2-s + (−0.240 + 0.138i)3-s + (−0.971 − 0.238i)4-s + (0.714 − 1.23i)5-s + (0.108 + 0.255i)6-s + (0.262 + 0.964i)7-s + (−0.353 + 0.935i)8-s + (−0.461 + 0.799i)9-s + (−1.14 − 0.857i)10-s + (0.241 + 0.418i)11-s + (0.266 − 0.0774i)12-s − 0.385·13-s + (0.989 − 0.144i)14-s + 0.396i·15-s + (0.886 + 0.463i)16-s + (0.845 − 0.488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.679297 - 0.500424i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.679297 - 0.500424i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.169 + 1.40i)T \) |
| 7 | \( 1 + (-0.694 - 2.55i)T \) |
good | 3 | \( 1 + (0.416 - 0.240i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.59 + 2.76i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.800 - 1.38i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 17 | \( 1 + (-3.48 + 2.01i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.56 + 2.63i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.83 + 2.21i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.10iT - 29T^{2} \) |
| 31 | \( 1 + (0.0579 + 0.100i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.63 + 2.67i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.21iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (-5.05 + 8.76i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.13 + 3.54i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.38 - 2.53i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.21 - 7.29i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.01 - 8.69i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.29iT - 71T^{2} \) |
| 73 | \( 1 + (-9.30 + 5.37i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.3 - 5.96i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 14.9iT - 83T^{2} \) |
| 89 | \( 1 + (-1.5 - 0.866i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2.87iT - 97T^{2} \) |
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
−14.78446360208411867330720473170, −13.66817660180281782305695386624, −12.56942797711840506829046901143, −11.87410335070880401719929512203, −10.45007910362127310451250615853, −9.254000141675969425264204001226, −8.409918306769938684211647501948, −5.57637555101040971646264820839, −4.78194298745032568164569571545, −2.15983832251087429689402678176,
3.71851100473250361583582482992, 5.88071359817964704063112862272, 6.69475817690909313554539557738, 7.977218556795374272567443228196, 9.673751035589436658578175764006, 10.70554870643721000316842646601, 12.31260370706290523786878010874, 13.86181221046588967478742979815, 14.32231904970114104722852578902, 15.26471919884032658692676230840