L(s) = 1 | + (−0.125 − 1.99i)2-s + (1.22 − 1.22i)3-s + (−3.96 + 0.500i)4-s + (3.32 − 3.32i)5-s + (−2.59 − 2.29i)6-s − 4.04·7-s + (1.49 + 7.85i)8-s − 2.99i·9-s + (−7.05 − 6.22i)10-s + (6.82 + 6.82i)11-s + (−4.24 + 5.47i)12-s + (4.29 + 4.29i)13-s + (0.506 + 8.06i)14-s − 8.14i·15-s + (15.4 − 3.97i)16-s + 30.1·17-s + ⋯ |
L(s) = 1 | + (−0.0626 − 0.998i)2-s + (0.408 − 0.408i)3-s + (−0.992 + 0.125i)4-s + (0.665 − 0.665i)5-s + (−0.433 − 0.381i)6-s − 0.577·7-s + (0.187 + 0.982i)8-s − 0.333i·9-s + (−0.705 − 0.622i)10-s + (0.620 + 0.620i)11-s + (−0.353 + 0.456i)12-s + (0.330 + 0.330i)13-s + (0.0361 + 0.576i)14-s − 0.543i·15-s + (0.968 − 0.248i)16-s + 1.77·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.771356 - 0.889311i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.771356 - 0.889311i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.125 + 1.99i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-3.32 + 3.32i)T - 25iT^{2} \) |
| 7 | \( 1 + 4.04T + 49T^{2} \) |
| 11 | \( 1 + (-6.82 - 6.82i)T + 121iT^{2} \) |
| 13 | \( 1 + (-4.29 - 4.29i)T + 169iT^{2} \) |
| 17 | \( 1 - 30.1T + 289T^{2} \) |
| 19 | \( 1 + (19.7 - 19.7i)T - 361iT^{2} \) |
| 23 | \( 1 + 28.2T + 529T^{2} \) |
| 29 | \( 1 + (21.3 + 21.3i)T + 841iT^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 + (42.8 - 42.8i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 48.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-32.6 - 32.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 15.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (0.476 - 0.476i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-9.97 - 9.97i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-37.9 - 37.9i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-20.0 + 20.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 40.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 30.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 130. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (2.26 - 2.26i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 72.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 112.T + 9.40e3T^{2} \) |
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
−14.67072726491174516786435011415, −13.70117321108920930880684165015, −12.68489516092536826282926810767, −11.95713945366780694678000460527, −10.07921593900380815152938618213, −9.362556412246055969598063405800, −8.018242416930068133620961705904, −5.88035740005531551202570862876, −3.85987524989755973398501719249, −1.74188252212475400268684983584,
3.54826847181597847311203393806, 5.64177409392442631817642587643, 6.80831588622261272543734793293, 8.401838176818770097639112220975, 9.590278315726681588219066947751, 10.56702436935985127800263921617, 12.68170440624641657090595952197, 14.01215264998878059901163350957, 14.48046588580079961057182034693, 15.77987812980426514565064547600