L(s) = 1 | + (0.759 − 1.19i)2-s − 1.39i·3-s + (−0.846 − 1.81i)4-s + (0.535 + 2.17i)5-s + (−1.66 − 1.05i)6-s + (−2.13 + 2.13i)7-s + (−2.80 − 0.366i)8-s + 1.05·9-s + (2.99 + 1.01i)10-s + (2.17 − 2.17i)11-s + (−2.52 + 1.17i)12-s + 1.54·13-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 | + (0.536 − 0.843i)2-s − 0.804i·3-s + (−0.423 − 0.905i)4-s + (0.239 + 0.970i)5-s + (−0.678 − 0.431i)6-s + (−0.806 + 0.806i)7-s + (−0.991 − 0.129i)8-s + 0.353·9-s + (0.947 + 0.319i)10-s + (0.654 − 0.654i)11-s + (−0.728 + 0.340i)12-s + 0.428·13-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.882974 - 0.744610i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.882974 - 0.744610i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.759 + 1.19i)T \) |
| 5 | \( 1 + (-0.535 - 2.17i)T \) |
good | 3 | \( 1 + 1.39iT - 3T^{2} \) |
| 7 | \( 1 + (2.13 - 2.13i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.17 + 2.17i)T - 11iT^{2} \) |
| 13 | \( 1 - 1.54T + 13T^{2} \) |
| 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 | \( 1 - 7.24T + 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 2.02T + 43T^{2} \) |
| 47 | \( 1 + (3.34 + 3.34i)T + 47iT^{2} \) |
| 53 | \( 1 + 7.30iT - 53T^{2} \) |
| 59 | \( 1 + (-3.52 - 3.52i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1.41 + 1.41i)T - 61iT^{2} \) |
| 67 | \( 1 - 0.748T + 67T^{2} \) |
| 71 | \( 1 + 0.269T + 71T^{2} \) |
| 73 | \( 1 + (0.811 - 0.811i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.80T + 79T^{2} \) |
| 83 | \( 1 - 12.8iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + (-6.33 + 6.33i)T - 97iT^{2} \) |
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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
−13.84714049698878547553928508879, −13.02967275712517779760318348648, −12.13903424145145382927399247252, −11.05188722666484513474238579127, −9.972568622287418811411631121407, −8.697024876674212192017550470070, −6.64615192626781552104619187530, −6.02548392547115128554610123314, −3.71059470283862852366981585869, −2.17416218862061596664810628327,
3.90143342478921247978563678283, 4.71880777950348626402217786536, 6.29630038788039366025039838134, 7.57205385146799323431187689089, 9.221806109058678684740571154958, 9.740714741269057855593157499125, 11.56854884245416954127529104293, 12.99069032891268509814796183583, 13.42517302196962976804928287427, 14.79630075931568612889052480062