L(s) = 1 | + 0.981i·2-s + 1.03·4-s + (0.940 + 1.62i)5-s + 2.98i·8-s + (−1.59 + 0.923i)10-s + (3.54 + 2.04i)11-s + (3.51 + 2.02i)13-s − 0.852·16-s + (0.810 + 1.40i)17-s + (−7.03 − 4.06i)19-s + (0.974 + 1.68i)20-s + (−2.00 + 3.47i)22-s + (3.73 − 2.15i)23-s + (0.730 − 1.26i)25-s + (−1.99 + 3.44i)26-s + ⋯ |
L(s) = 1 | + 0.694i·2-s + 0.518·4-s + (0.420 + 0.728i)5-s + 1.05i·8-s + (−0.505 + 0.291i)10-s + (1.06 + 0.616i)11-s + (0.974 + 0.562i)13-s − 0.213·16-s + (0.196 + 0.340i)17-s + (−1.61 − 0.932i)19-s + (0.217 + 0.377i)20-s + (−0.427 + 0.740i)22-s + (0.778 − 0.449i)23-s + (0.146 − 0.253i)25-s + (−0.390 + 0.676i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.282 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.327249479\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.327249479\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.981iT - 2T^{2} \) |
| 5 | \( 1 + (-0.940 - 1.62i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.54 - 2.04i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.51 - 2.02i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.810 - 1.40i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (7.03 + 4.06i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.73 + 2.15i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.542 + 0.313i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.27iT - 31T^{2} \) |
| 37 | \( 1 + (3.97 - 6.87i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.912 - 1.57i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.53 + 6.12i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 7.93T + 47T^{2} \) |
| 53 | \( 1 + (-7.24 + 4.18i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.17T + 59T^{2} \) |
| 61 | \( 1 - 3.74iT - 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 14.4iT - 71T^{2} \) |
| 73 | \( 1 + (-3.28 + 1.89i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 8.36T + 79T^{2} \) |
| 83 | \( 1 + (4.38 + 7.59i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.90 - 8.49i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.4 + 6.61i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.897570358010782025406672690511, −8.797395776946520805895166374890, −8.323900356552833354709997922945, −6.98268487593331994494108031579, −6.64447683420575329319334496750, −6.17104959912880435744422463576, −4.92299330243359128706348536041, −3.88395883799872034043386489054, −2.61184506556384665195907887033, −1.68094197948728875558023645304,
1.04225815382723315226100752628, 1.82396282255830445906770831372, 3.24916292555792393094547615806, 3.89702083756256987197202823039, 5.19185816614550280108878087865, 6.16127057368969609724433231469, 6.70170733358340776080793661647, 7.934148352683679709335070476945, 8.806746094827743058785429614694, 9.354977429250132787950319956374