L(s) = 1 | + (−1.22 − 1.22i)3-s + (0.909 + 0.909i)5-s + 0.654·7-s + 2.99i·9-s + (13.3 − 13.3i)11-s + (8.32 − 8.32i)13-s − 2.22i·15-s − 3.93·17-s + (−16.8 − 16.8i)19-s + (−0.801 − 0.801i)21-s + 23.1·23-s − 23.3i·25-s + (3.67 − 3.67i)27-s + (35.6 − 35.6i)29-s + 45.5i·31-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.181 + 0.181i)5-s + 0.0935·7-s + 0.333i·9-s + (1.21 − 1.21i)11-s + (0.640 − 0.640i)13-s − 0.148i·15-s − 0.231·17-s + (−0.889 − 0.889i)19-s + (−0.0381 − 0.0381i)21-s + 1.00·23-s − 0.933i·25-s + (0.136 − 0.136i)27-s + (1.22 − 1.22i)29-s + 1.46i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.24998 - 0.654568i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24998 - 0.654568i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
good | 5 | \( 1 + (-0.909 - 0.909i)T + 25iT^{2} \) |
| 7 | \( 1 - 0.654T + 49T^{2} \) |
| 11 | \( 1 + (-13.3 + 13.3i)T - 121iT^{2} \) |
| 13 | \( 1 + (-8.32 + 8.32i)T - 169iT^{2} \) |
| 17 | \( 1 + 3.93T + 289T^{2} \) |
| 19 | \( 1 + (16.8 + 16.8i)T + 361iT^{2} \) |
| 23 | \( 1 - 23.1T + 529T^{2} \) |
| 29 | \( 1 + (-35.6 + 35.6i)T - 841iT^{2} \) |
| 31 | \( 1 - 45.5iT - 961T^{2} \) |
| 37 | \( 1 + (-10.1 - 10.1i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 28.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (22.7 - 22.7i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 10.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-41.5 - 41.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-21.0 + 21.0i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (68.7 - 68.7i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (67.8 + 67.8i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 33.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 18.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.29iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-72.0 - 72.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 10.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 143.T + 9.40e3T^{2} \) |
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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
−12.05313428872087786830448225152, −11.20164853397477906397488954023, −10.45029309957100921357030776679, −8.976706973194079817440356422957, −8.216115269877501449593733891294, −6.64621779271117959408007360640, −6.12415670183239467039815755246, −4.59460777317616314944557983637, −2.98848440125084608347592057126, −0.994332729000749753185258906975,
1.64485216200630904756813040301, 3.82572984551057280577737066098, 4.81474735482928733396929625419, 6.21457770057744251420834720243, 7.10769655466420844712566315217, 8.711504200373937837769183283038, 9.457354163359421793246297895315, 10.50476846576218431927289618142, 11.51086201760934106886953477592, 12.32674506311227112715724143369