L(s) = 1 | + (−0.546 − 1.30i)2-s + (−0.5 − 0.866i)3-s + (−1.40 + 1.42i)4-s + (−3.33 − 1.92i)5-s + (−0.856 + 1.12i)6-s + (1.59 − 2.11i)7-s + (2.62 + 1.05i)8-s + (−0.499 + 0.866i)9-s + (−0.690 + 5.40i)10-s + (1.17 − 0.681i)11-s + (1.93 + 0.502i)12-s − 0.369i·13-s + (−3.62 − 0.923i)14-s + 3.85i·15-s + (−0.0640 − 3.99i)16-s + (3.89 − 2.25i)17-s + ⋯ |
L(s) = 1 | + (−0.386 − 0.922i)2-s + (−0.288 − 0.499i)3-s + (−0.701 + 0.712i)4-s + (−1.49 − 0.862i)5-s + (−0.349 + 0.459i)6-s + (0.602 − 0.798i)7-s + (0.928 + 0.371i)8-s + (−0.166 + 0.288i)9-s + (−0.218 + 1.71i)10-s + (0.355 − 0.205i)11-s + (0.558 + 0.144i)12-s − 0.102i·13-s + (−0.969 − 0.246i)14-s + 0.995i·15-s + (−0.0160 − 0.999i)16-s + (0.945 − 0.545i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.151847 - 0.544214i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.151847 - 0.544214i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.546 + 1.30i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.59 + 2.11i)T \) |
good | 5 | \( 1 + (3.33 + 1.92i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.17 + 0.681i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 0.369iT - 13T^{2} \) |
| 17 | \( 1 + (-3.89 + 2.25i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0330 + 0.0573i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.77 - 1.60i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 3.11T + 29T^{2} \) |
| 31 | \( 1 + (3.01 + 5.22i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.74 + 4.75i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8.45iT - 41T^{2} \) |
| 43 | \( 1 - 6.30iT - 43T^{2} \) |
| 47 | \( 1 + (0.712 - 1.23i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.27 - 2.20i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.71 - 2.97i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.23 - 0.715i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.45 + 4.88i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.9iT - 71T^{2} \) |
| 73 | \( 1 + (1.56 - 0.900i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.8 - 6.24i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + (-1.11 - 0.646i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.88iT - 97T^{2} \) |
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
−13.42134220261031249466325130068, −12.49424903669101243615606874819, −11.58136610346417553228058265542, −11.02834662333599580496100942484, −9.365270665827275976532390798742, −8.026593980869380322179753172266, −7.51224669935460926617653762445, −4.90177724160885104452297277009, −3.70539036095209240374571888226, −0.938493665518165874507254169903,
3.82782947239650031029925458891, 5.26861307111125023510256063710, 6.78208210969383974135464162230, 7.88687843598501902043543816548, 8.881855270378190306077556638528, 10.36954200874338805258411346892, 11.33429759334208263186728079013, 12.35919868597485380749940600314, 14.37971141887090676452656571127, 14.93274108519178507955075462586