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 | \( 1 + 112.T + 9.40e3T^{2} \) |
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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
−15.77987812980426514565064547600, −14.48046588580079961057182034693, −14.01215264998878059901163350957, −12.68170440624641657090595952197, −10.56702436935985127800263921617, −9.590278315726681588219066947751, −8.401838176818770097639112220975, −6.80831588622261272543734793293, −5.64177409392442631817642587643, −3.54826847181597847311203393806,
1.74188252212475400268684983584, 3.85987524989755973398501719249, 5.88035740005531551202570862876, 8.018242416930068133620961705904, 9.362556412246055969598063405800, 10.07921593900380815152938618213, 11.95713945366780694678000460527, 12.68489516092536826282926810767, 13.70117321108920930880684165015, 14.67072726491174516786435011415