L(s) = 1 | + (−1.16 − 1.62i)2-s + (−1.30 + 3.78i)4-s − 7.67·5-s − 7.21·7-s + (7.67 − 2.27i)8-s + (8.90 + 12.4i)10-s − 6.05·11-s + 2.29i·13-s + (8.37 + 11.7i)14-s + (−12.6 − 9.85i)16-s + 21.8i·17-s − 34.8i·19-s + (9.99 − 29.0i)20-s + (7.02 + 9.85i)22-s − 21.5i·23-s + ⋯ |
L(s) = 1 | + (−0.580 − 0.814i)2-s + (−0.325 + 0.945i)4-s − 1.53·5-s − 1.03·7-s + (0.958 − 0.283i)8-s + (0.890 + 1.24i)10-s − 0.550·11-s + 0.176i·13-s + (0.598 + 0.838i)14-s + (−0.787 − 0.615i)16-s + 1.28i·17-s − 1.83i·19-s + (0.499 − 1.45i)20-s + (0.319 + 0.447i)22-s − 0.935i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00905959 + 0.0261278i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00905959 + 0.0261278i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.16 + 1.62i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 7.67T + 25T^{2} \) |
| 7 | \( 1 + 7.21T + 49T^{2} \) |
| 11 | \( 1 + 6.05T + 121T^{2} \) |
| 13 | \( 1 - 2.29iT - 169T^{2} \) |
| 17 | \( 1 - 21.8iT - 289T^{2} \) |
| 19 | \( 1 + 34.8iT - 361T^{2} \) |
| 23 | \( 1 + 21.5iT - 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 + 37.6T + 961T^{2} \) |
| 37 | \( 1 - 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 13.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 60.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 3.34iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 35.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 37.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 25.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 25.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 37.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 77.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 31.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 55.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 5.43iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 52.8T + 9.40e3T^{2} \) |
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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
−13.18393334689880668538886115996, −12.54077464393041173046968603983, −11.39707222244309191844831768123, −10.54906178137271793031607808836, −9.138263954711513727228380677398, −8.079610595082579360275236750859, −6.91246096114374111982788961004, −4.34129296725089818607756519613, −3.07227801042497684303982487122, −0.02820736993504295107020202610,
3.72856895031028334834369935121, 5.53972873926725916349702365393, 7.16599370116573791676428071980, 7.87654863736013472150615162962, 9.217201773024487306885050332457, 10.39435835962985762848731304125, 11.68425001432511841431693917678, 12.90785006087423576072736670417, 14.28078210862211718756466381512, 15.43426083823280538050339183775