L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.866 + 1.5i)5-s + (−0.5 + 0.866i)7-s − 0.999·8-s + 1.73·10-s + (0.633 − 1.09i)11-s + (0.5 + 0.866i)13-s + (0.499 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 0.464·17-s + 4.19·19-s + (0.866 − 1.49i)20-s + (−0.633 − 1.09i)22-s + (2.36 + 4.09i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.387 + 0.670i)5-s + (−0.188 + 0.327i)7-s − 0.353·8-s + 0.547·10-s + (0.191 − 0.331i)11-s + (0.138 + 0.240i)13-s + (0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + 0.112·17-s + 0.962·19-s + (0.193 − 0.335i)20-s + (−0.135 − 0.234i)22-s + (0.493 + 0.854i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.061587592\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.061587592\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.866 - 1.5i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.633 + 1.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 0.464T + 17T^{2} \) |
| 19 | \( 1 - 4.19T + 19T^{2} \) |
| 23 | \( 1 + (-2.36 - 4.09i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.232 - 0.401i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.09 - 5.36i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + (-4.73 - 8.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.19 + 7.26i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.09 - 7.09i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + (-1.09 - 1.90i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.69 + 9.86i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.09 - 5.36i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 1.19T + 73T^{2} \) |
| 79 | \( 1 + (2.09 - 3.63i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.36 + 4.09i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 5.53T + 89T^{2} \) |
| 97 | \( 1 + (-8 + 13.8i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.786990835244113295150307400637, −9.297095718774881250373040855388, −8.256579682408191220862889909468, −7.16721749755116338513002090222, −6.28352584056085649683846085658, −5.56015334720838172277636743978, −4.51264745035404419538481154228, −3.31126729417082316705035204194, −2.66286898990005208061405875466, −1.27311627212400670828822401088,
0.978629771851916526419799553484, 2.64222264548782763048959330132, 3.88585515206760635598243011174, 4.77043539528773739842918602460, 5.58294062194433010786551089133, 6.41733918863507800066994846354, 7.34689826559342750035329569971, 8.063993408619604779634380676563, 9.069142697683795720035972983858, 9.590275993817302598642515402644