L(s) = 1 | + 1.73i·2-s − 0.999·4-s + (−1.5 + 2.59i)5-s + 1.73i·8-s + (−4.5 − 2.59i)10-s + (−1.5 + 0.866i)11-s + (−1.5 + 0.866i)13-s − 5·16-s + (−1.5 + 2.59i)17-s + (4.5 − 2.59i)19-s + (1.49 − 2.59i)20-s + (−1.49 − 2.59i)22-s + (−4.5 − 2.59i)23-s + (−2 − 3.46i)25-s + (−1.49 − 2.59i)26-s + ⋯ |
L(s) = 1 | + 1.22i·2-s − 0.499·4-s + (−0.670 + 1.16i)5-s + 0.612i·8-s + (−1.42 − 0.821i)10-s + (−0.452 + 0.261i)11-s + (−0.416 + 0.240i)13-s − 1.25·16-s + (−0.363 + 0.630i)17-s + (1.03 − 0.596i)19-s + (0.335 − 0.580i)20-s + (−0.319 − 0.553i)22-s + (−0.938 − 0.541i)23-s + (−0.400 − 0.692i)25-s + (−0.294 − 0.509i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7710555876\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7710555876\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.5 + 2.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 + 2.59i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.5 - 2.59i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 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 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (7.5 + 4.33i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + (-4.5 - 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (-7.5 + 12.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.30807249642145749881106989024, −9.142918382446096905217017527247, −8.261749417963911313119269930079, −7.54022840388427810196141323217, −6.99129590201556492953195970126, −6.38185004607680493565648384153, −5.36953059002454430937301952311, −4.46357951635280739302406637157, −3.25825189226011746353617589519, −2.24223482028349407986479729627,
0.31432228350817491950959462870, 1.45511418391115584070295651270, 2.72755536263351577810185112601, 3.66969626122684306767588749446, 4.57652760072173003380963784578, 5.34135586073792486108463949430, 6.59133049012727989235319212468, 7.77844876444375770307166673329, 8.241252654712571118492266540321, 9.430952200025500711465995758466