| L(s) = 1 | + (−2.54 − 1.59i)3-s + (−4.96 + 0.625i)5-s + 5.93i·7-s + (3.92 + 8.10i)9-s − 5.99·11-s − 15.6·13-s + (13.6 + 6.31i)15-s + 14.2·17-s + 29.6i·19-s + (9.46 − 15.0i)21-s − 21.5·23-s + (24.2 − 6.20i)25-s + (2.93 − 26.8i)27-s + 18.1·29-s − 8.55·31-s + ⋯ |
| L(s) = 1 | + (−0.847 − 0.531i)3-s + (−0.992 + 0.125i)5-s + 0.848i·7-s + (0.435 + 0.900i)9-s − 0.545·11-s − 1.20·13-s + (0.907 + 0.420i)15-s + 0.836·17-s + 1.55i·19-s + (0.450 − 0.718i)21-s − 0.935·23-s + (0.968 − 0.248i)25-s + (0.108 − 0.994i)27-s + 0.625·29-s − 0.276·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3420139982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3420139982\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.54 + 1.59i)T \) |
| 5 | \( 1 + (4.96 - 0.625i)T \) |
| good | 7 | \( 1 - 5.93iT - 49T^{2} \) |
| 11 | \( 1 + 5.99T + 121T^{2} \) |
| 13 | \( 1 + 15.6T + 169T^{2} \) |
| 17 | \( 1 - 14.2T + 289T^{2} \) |
| 19 | \( 1 - 29.6iT - 361T^{2} \) |
| 23 | \( 1 + 21.5T + 529T^{2} \) |
| 29 | \( 1 - 18.1T + 841T^{2} \) |
| 31 | \( 1 + 8.55T + 961T^{2} \) |
| 37 | \( 1 - 13.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 18.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 27.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 37.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 93.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 83.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 39.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 103.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 14.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 95.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 116. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.9iT - 9.40e3T^{2} \) |
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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
−9.936680563190022134938018158840, −8.427847496444266594732218215385, −7.85473383469818065121107491244, −7.17033895790017465432617285291, −6.06913131758415829923653621154, −5.34218417589055847652380384591, −4.42754603787375717334417854402, −3.09891825438038323839342721747, −1.85393169857599099022072101570, −0.17601550172714066998978538164,
0.802763432611459447643155136859, 2.88791315675597203785516818996, 4.06095439578475651786451702236, 4.70211566811199912713221998283, 5.52387507518449593946902177215, 6.87864719474440723576144666989, 7.35796202912576686291184782274, 8.297960005608342960250095099047, 9.466731794293834217434884293330, 10.18794799004406397425071904691
