L(s) = 1 | + (0.866 − 0.5i)2-s + (0.866 − 1.5i)3-s + (0.499 − 0.866i)4-s − 1.73i·6-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (−3 − 5.19i)11-s + (−0.866 − 1.49i)12-s + (1.73 + i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−2.59 − 1.5i)18-s + 4·19-s + 1.73i·21-s + (−5.19 − 3i)22-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.499 − 0.866i)3-s + (0.249 − 0.433i)4-s − 0.707i·6-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.904 − 1.56i)11-s + (−0.250 − 0.433i)12-s + (0.480 + 0.277i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.612 − 0.353i)18-s + 0.917·19-s + 0.377i·21-s + (−1.10 − 0.639i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26876 - 1.70110i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26876 - 1.70110i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.866 - 0.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-7.79 - 4.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.92 - 4i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.59 - 1.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.2 - 6.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.79 + 4.5i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.28412820835085851206095981237, −9.957216744369929183557897769699, −8.909106393785235283491203548228, −8.113591939538624957518198675340, −7.01372733932440211732738886874, −6.06521051160828070472999981437, −5.20009884125136178087571030752, −3.40296454368117312028997048122, −2.85997329212284921635452178448, −1.13358732924119300500904134295,
2.46728650356243913939619277202, 3.51809002106319628068327228091, 4.70552502006216433933829846652, 5.29694098179610701903627594050, 6.77308771332462156169603780472, 7.64103821042054999448358432711, 8.603911167433507993648733687964, 9.665066905862061700772069709335, 10.36880473355419121252972084433, 11.24421285267855315540868774852