| L(s) = 1 | + (−1.94 − 0.464i)2-s + (−2.50 − 1.64i)3-s + (3.56 + 1.80i)4-s + (−1.64 − 2.84i)5-s + (4.10 + 4.37i)6-s + (−4.94 + 8.56i)7-s + (−6.09 − 5.17i)8-s + (3.56 + 8.26i)9-s + (1.87 + 6.30i)10-s + (−7.26 + 12.5i)11-s + (−5.95 − 10.4i)12-s + (−8.15 + 4.71i)13-s + (13.5 − 14.3i)14-s + (−0.574 + 9.84i)15-s + (9.45 + 12.9i)16-s − 16.6i·17-s + ⋯ |
| L(s) = 1 | + (−0.972 − 0.232i)2-s + (−0.835 − 0.549i)3-s + (0.891 + 0.452i)4-s + (−0.328 − 0.569i)5-s + (0.684 + 0.728i)6-s + (−0.706 + 1.22i)7-s + (−0.762 − 0.647i)8-s + (0.395 + 0.918i)9-s + (0.187 + 0.630i)10-s + (−0.660 + 1.14i)11-s + (−0.496 − 0.867i)12-s + (−0.627 + 0.362i)13-s + (0.971 − 1.02i)14-s + (−0.0382 + 0.656i)15-s + (0.590 + 0.806i)16-s − 0.977i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0712628 + 0.131229i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0712628 + 0.131229i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.94 + 0.464i)T \) |
| 3 | \( 1 + (2.50 + 1.64i)T \) |
| good | 5 | \( 1 + (1.64 + 2.84i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (4.94 - 8.56i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (7.26 - 12.5i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (8.15 - 4.71i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 16.6iT - 289T^{2} \) |
| 19 | \( 1 + 11.8iT - 361T^{2} \) |
| 23 | \( 1 + (27.6 - 15.9i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (13.0 - 22.6i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (8.69 + 15.0i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 40.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (6.97 - 4.02i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (50.5 + 29.1i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-0.0690 - 0.0398i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 10.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-53.8 - 93.3i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (46.2 + 26.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-69.3 + 40.0i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 16.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 30.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-46.7 + 81.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (7.74 - 13.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 12.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (91.1 - 157. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−15.31581213078676268530686802380, −13.10792397606897372327860837859, −12.15323907458039309810209998157, −11.77535478157960807935595648956, −10.15542796466952198698190143790, −9.187222399329643856541196152709, −7.77765373186451491614044972396, −6.68591210479570557487890580377, −5.14820989443084986434421306184, −2.28558681585415749610262476339,
0.18194934358734528490279481045, 3.58775132901834145224856216200, 5.79541992799963929286565924567, 6.88743847920286876061018383610, 8.113642594672560662083646654326, 9.885718962955924344537020202988, 10.50024987708275183541796285519, 11.26009201143602684589167152294, 12.70002394190144052734255078056, 14.35609602604620512127493096533
