L(s) = 1 | + (1.19 + 0.759i)2-s − 1.39·3-s + (0.846 + 1.81i)4-s + (2.17 + 0.535i)5-s + (−1.66 − 1.05i)6-s + (−2.13 − 2.13i)7-s + (−0.366 + 2.80i)8-s − 1.05·9-s + (2.18 + 2.28i)10-s + (2.17 − 2.17i)11-s + (−1.17 − 2.52i)12-s − 1.54i·13-s + (−0.925 − 4.16i)14-s + (−3.02 − 0.745i)15-s + (−2.56 + 3.06i)16-s + (−3.86 − 3.86i)17-s + ⋯ |
L(s) = 1 | + (0.843 + 0.536i)2-s − 0.804·3-s + (0.423 + 0.905i)4-s + (0.970 + 0.239i)5-s + (−0.678 − 0.431i)6-s + (−0.806 − 0.806i)7-s + (−0.129 + 0.991i)8-s − 0.353·9-s + (0.690 + 0.723i)10-s + (0.654 − 0.654i)11-s + (−0.340 − 0.728i)12-s − 0.428i·13-s + (−0.247 − 1.11i)14-s + (−0.780 − 0.192i)15-s + (−0.641 + 0.766i)16-s + (−0.937 − 0.937i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11420 + 0.466648i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11420 + 0.466648i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.19 - 0.759i)T \) |
| 5 | \( 1 + (-2.17 - 0.535i)T \) |
good | 3 | \( 1 + 1.39T + 3T^{2} \) |
| 7 | \( 1 + (2.13 + 2.13i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.17 + 2.17i)T - 11iT^{2} \) |
| 13 | \( 1 + 1.54iT - 13T^{2} \) |
| 17 | \( 1 + (3.86 + 3.86i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.0136 + 0.0136i)T - 19iT^{2} \) |
| 23 | \( 1 + (3.15 - 3.15i)T - 23iT^{2} \) |
| 29 | \( 1 + (-3.33 - 3.33i)T + 29iT^{2} \) |
| 31 | \( 1 - 8.92iT - 31T^{2} \) |
| 37 | \( 1 - 7.24iT - 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 2.02iT - 43T^{2} \) |
| 47 | \( 1 + (-3.34 + 3.34i)T - 47iT^{2} \) |
| 53 | \( 1 + 7.30T + 53T^{2} \) |
| 59 | \( 1 + (3.52 + 3.52i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1.41 + 1.41i)T - 61iT^{2} \) |
| 67 | \( 1 - 0.748iT - 67T^{2} \) |
| 71 | \( 1 + 0.269T + 71T^{2} \) |
| 73 | \( 1 + (-0.811 - 0.811i)T + 73iT^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + (-6.33 - 6.33i)T + 97iT^{2} \) |
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
−14.08173438236835268881391626920, −13.76399410503305065683477059451, −12.56028944311603290643465983667, −11.40355214231361299646869914670, −10.38625216647679782215687597257, −8.847149066992115258971924757319, −6.93366952907401415154724324489, −6.24556113042840509821568501081, −5.09512751218235227888930473354, −3.22407160541480759212436266856,
2.31173427755191507676879502342, 4.48544466313376896212931560133, 6.03295846021385348716602200187, 6.32734776673178668006969255201, 9.073244535387961587336375269419, 10.00809273600621581545024318620, 11.22657351593398758029829122752, 12.25810282320259094963904962990, 12.92838222779565633467366668712, 14.09839488534551599229865464798