L(s) = 1 | − 1.73i·3-s + (−1.5 − 2.59i)5-s + (−0.5 + 0.866i)7-s − 2.99·9-s + (1.5 − 2.59i)11-s + (0.5 + 0.866i)13-s + (−4.5 + 2.59i)15-s + 6·17-s + 4·19-s + (1.49 + 0.866i)21-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s + 5.19i·27-s + (−1.5 + 2.59i)29-s + (2.5 + 4.33i)31-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + (−0.670 − 1.16i)5-s + (−0.188 + 0.327i)7-s − 0.999·9-s + (0.452 − 0.783i)11-s + (0.138 + 0.240i)13-s + (−1.16 + 0.670i)15-s + 1.45·17-s + 0.917·19-s + (0.327 + 0.188i)21-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s + 0.999i·27-s + (−0.278 + 0.482i)29-s + (0.449 + 0.777i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.620791 - 0.739830i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.620791 - 0.739830i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
good | 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (5.5 - 9.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.49753018174528950969373641326, −12.17938940567116157236683203657, −11.17521245136674454721478943905, −9.419095640760460642674684366510, −8.451398717742178608733058238491, −7.71215399010766595203941265039, −6.27944521460285198715481266591, −5.13111030909990043887555761680, −3.34725078321806087360246920646, −1.10309822942172362190392132849,
3.12456873246481322824104634200, 4.04531457728349794313523609829, 5.62145106335824226285795652209, 7.06442974047196561827728526342, 8.037768318414551450244332052876, 9.686296243206800738249221035342, 10.17307402423722297079296465044, 11.35052065175557841697075784163, 11.98128141849072251647111542140, 13.63939936948839066671965589658