L(s) = 1 | + (−1.73 + 2.44i)3-s − 2.82i·5-s + 10.3·7-s + (−2.99 − 8.48i)9-s + 14.6i·11-s + 6·13-s + (6.92 + 4.89i)15-s + 22.6i·17-s + 10.3·19-s + (−18 + 25.4i)21-s + 29.3i·23-s + 17·25-s + (25.9 + 7.34i)27-s − 31.1i·29-s − 31.1·31-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s − 0.565i·5-s + 1.48·7-s + (−0.333 − 0.942i)9-s + 1.33i·11-s + 0.461·13-s + (0.461 + 0.326i)15-s + 1.33i·17-s + 0.546·19-s + (−0.857 + 1.21i)21-s + 1.27i·23-s + 0.680·25-s + (0.962 + 0.272i)27-s − 1.07i·29-s − 1.00·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.27022 + 0.657516i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27022 + 0.657516i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 - 2.44i)T \) |
good | 5 | \( 1 + 2.82iT - 25T^{2} \) |
| 7 | \( 1 - 10.3T + 49T^{2} \) |
| 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 - 6T + 169T^{2} \) |
| 17 | \( 1 - 22.6iT - 289T^{2} \) |
| 19 | \( 1 - 10.3T + 361T^{2} \) |
| 23 | \( 1 - 29.3iT - 529T^{2} \) |
| 29 | \( 1 + 31.1iT - 841T^{2} \) |
| 31 | \( 1 + 31.1T + 961T^{2} \) |
| 37 | \( 1 - 38T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.65iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 14.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 14.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 22T + 3.72e3T^{2} \) |
| 67 | \( 1 + 114.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 29.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 30T + 5.32e3T^{2} \) |
| 79 | \( 1 + 31.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 73.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.65iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 90T + 9.40e3T^{2} \) |
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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
−12.20244576266472459658654821740, −11.45296855248256347179285067826, −10.57408083106341655455679315324, −9.556183032904937557900245032665, −8.535448451387586839593442363822, −7.44175615020804701800707046341, −5.80458253610932666629022668597, −4.86792930215584232727671835704, −4.00362511082225894223277908158, −1.57108700097724275686614189191,
1.06871512151281810989255269874, 2.80087441125730779139323786691, 4.81896274423511660699511031808, 5.84481953356596687067487609140, 7.02679106083668334129796750170, 7.954762290277353143503316656915, 8.890019075588946499470974626715, 10.81295518720088555282981347994, 11.09140128726996742402026178701, 11.94139043721492925234238407346