| L(s) = 1 | + (2.37 − 0.866i)2-s + (−0.113 + 0.642i)3-s + (3.37 − 2.83i)4-s + (−1.03 − 0.866i)5-s + (0.286 + 1.62i)6-s + (−0.766 − 1.32i)7-s + (3.05 − 5.28i)8-s + (2.41 + 0.880i)9-s + (−3.20 − 1.16i)10-s + (0.592 − 1.02i)11-s + (1.43 + 2.49i)12-s + (0.471 + 2.67i)13-s + (−2.97 − 2.49i)14-s + (0.673 − 0.565i)15-s + (1.15 − 6.53i)16-s + (−3.64 + 1.32i)17-s + ⋯ |
| L(s) = 1 | + (1.68 − 0.612i)2-s + (−0.0654 + 0.371i)3-s + (1.68 − 1.41i)4-s + (−0.461 − 0.387i)5-s + (0.117 + 0.664i)6-s + (−0.289 − 0.501i)7-s + (1.07 − 1.86i)8-s + (0.806 + 0.293i)9-s + (−1.01 − 0.368i)10-s + (0.178 − 0.309i)11-s + (0.415 + 0.719i)12-s + (0.130 + 0.742i)13-s + (−0.794 − 0.666i)14-s + (0.173 − 0.145i)15-s + (0.288 − 1.63i)16-s + (−0.884 + 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.76525 - 1.38175i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.76525 - 1.38175i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-2.37 + 0.866i)T + (1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (0.113 - 0.642i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (1.03 + 0.866i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (0.766 + 1.32i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.592 + 1.02i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.471 - 2.67i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.64 - 1.32i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (3.87 - 3.25i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-4.37 - 1.59i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.91 - 3.32i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 + (1.73 - 9.83i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (6.66 + 5.59i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (0.539 + 0.196i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-2.25 + 1.89i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.69 + 1.34i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (3.46 - 2.90i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.65 - 1.32i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (5.31 + 4.45i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.06 + 6.03i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.70 + 9.65i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (6.15 + 10.6i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.421 + 2.38i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (6.92 - 2.52i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.62226155310165488384046244047, −10.61735665924386428955049678865, −9.973775109898204807786608587669, −8.541713199436767449462046852220, −7.06155272408284334387313737693, −6.25201268874183161977803209055, −4.88425513517925634903157993464, −4.26579434736974276958868896080, −3.44667010849219289029932484472, −1.76121628343968130038496707340,
2.44566335777261091915037054167, 3.67035848280779182973914119948, 4.58595991111283794719373169236, 5.78956151395153410023383275614, 6.67075548290558855457029970232, 7.28703339327048182324284799345, 8.353376936744597698306781632448, 9.874478796857193563046768896534, 11.11355931367262883861345247331, 12.06032886302530595320093411691
