L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s − 6.54·5-s + 1.31i·7-s − 7.99·8-s + (−6.54 + 11.3i)10-s − 4.30i·11-s + 0.824·13-s + (2.27 + 1.31i)14-s + (−8 + 13.8i)16-s − 0.274·17-s − 4.35i·19-s + (13.0 + 22.6i)20-s + (−7.45 − 4.30i)22-s + 33.5i·23-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 1.30·5-s + 0.187i·7-s − 0.999·8-s + (−0.654 + 1.13i)10-s − 0.391i·11-s + 0.0634·13-s + (0.162 + 0.0938i)14-s + (−0.5 + 0.866i)16-s − 0.0161·17-s − 0.229i·19-s + (0.654 + 1.13i)20-s + (−0.338 − 0.195i)22-s + 1.45i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8945880103\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8945880103\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 5 | \( 1 + 6.54T + 25T^{2} \) |
| 7 | \( 1 - 1.31iT - 49T^{2} \) |
| 11 | \( 1 + 4.30iT - 121T^{2} \) |
| 13 | \( 1 - 0.824T + 169T^{2} \) |
| 17 | \( 1 + 0.274T + 289T^{2} \) |
| 23 | \( 1 - 33.5iT - 529T^{2} \) |
| 29 | \( 1 - 33.3T + 841T^{2} \) |
| 31 | \( 1 - 48.7iT - 961T^{2} \) |
| 37 | \( 1 + 36.1T + 1.36e3T^{2} \) |
| 41 | \( 1 - 68.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 55.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 24.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 25.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 46.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 47.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 112. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 70.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 58.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 148. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 57.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 117.T + 9.40e3T^{2} \) |
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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
−10.72463598682992904680880741315, −9.575078386842523105763488896260, −8.727257486185417387283940750010, −7.86446528330975048505767387635, −6.75907520953996442052696039154, −5.58388057631319899709419517811, −4.61114082234618948465503408327, −3.68485962842146468818409222208, −2.86931917990618438288275324436, −1.18483661986862817620087055530,
0.31099568484085751872972548387, 2.76748814257069625423988500196, 4.05376222937524809672416149904, 4.45761577563932903451926374894, 5.75113670338850250257318826924, 6.79272740512043294036482143857, 7.52745564915086837333307893893, 8.225142657571714161248822434185, 8.987008343582397546874834777021, 10.21321997572908248390105890370