| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 1.55i·5-s + 12.3·7-s − 2.82i·8-s − 2.19·10-s + 14.6i·11-s − 10.8·13-s + 17.5i·14-s + 4.00·16-s + 28.9i·17-s + 3.60·19-s − 3.10i·20-s − 20.7·22-s − 14.6i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.310i·5-s + 1.77·7-s − 0.353i·8-s − 0.219·10-s + 1.33i·11-s − 0.831·13-s + 1.25i·14-s + 0.250·16-s + 1.70i·17-s + 0.189·19-s − 0.155i·20-s − 0.944·22-s − 0.638i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09891 + 1.09891i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09891 + 1.09891i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.55iT - 25T^{2} \) |
| 7 | \( 1 - 12.3T + 49T^{2} \) |
| 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 + 10.8T + 169T^{2} \) |
| 17 | \( 1 - 28.9iT - 289T^{2} \) |
| 19 | \( 1 - 3.60T + 361T^{2} \) |
| 23 | \( 1 + 14.6iT - 529T^{2} \) |
| 29 | \( 1 + 28.1iT - 841T^{2} \) |
| 31 | \( 1 - 8T + 961T^{2} \) |
| 37 | \( 1 - 22.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 25.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 16.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 84.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 91.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13T + 3.72e3T^{2} \) |
| 67 | \( 1 + 41.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 16.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 71.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 46.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 15.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 78.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 91.1T + 9.40e3T^{2} \) |
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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
−12.89165719914046000139989002731, −11.97113825818170461211529781036, −10.79787237351582273744794597635, −9.846473497852346163986420320802, −8.411301001047979272813089975474, −7.73411137293345524284399354022, −6.64288782676863942523476843447, −5.13477280980763541948708370669, −4.31801430535720735154613493551, −1.94274724425576045271595004294,
1.13041756349053652121415970419, 2.86487576909049947862593703507, 4.65771532530577310090220744271, 5.37867930653519245751032729566, 7.38556907458689586132010136765, 8.409774595413748179119831159457, 9.280163248393608496049934331649, 10.65445508643269214074624345903, 11.47367488945750550745271304388, 12.01830112138569347982127989533
