| L(s) = 1 | + (−1.66 + 1.66i)5-s − 13.3i·7-s + (−7.81 + 7.81i)11-s + (10.4 − 10.4i)13-s − 10.0i·17-s + (5.07 − 5.07i)19-s − 29.7·23-s + 19.4i·25-s + (18.6 + 18.6i)29-s − 27.1·31-s + (22.1 + 22.1i)35-s + (−13.3 − 13.3i)37-s + 34.9·41-s + (7.29 + 7.29i)43-s − 51.5i·47-s + ⋯ |
| L(s) = 1 | + (−0.332 + 0.332i)5-s − 1.90i·7-s + (−0.710 + 0.710i)11-s + (0.807 − 0.807i)13-s − 0.590i·17-s + (0.266 − 0.266i)19-s − 1.29·23-s + 0.778i·25-s + (0.644 + 0.644i)29-s − 0.875·31-s + (0.633 + 0.633i)35-s + (−0.360 − 0.360i)37-s + 0.852·41-s + (0.169 + 0.169i)43-s − 1.09i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3989732284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3989732284\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (1.66 - 1.66i)T - 25iT^{2} \) |
| 7 | \( 1 + 13.3iT - 49T^{2} \) |
| 11 | \( 1 + (7.81 - 7.81i)T - 121iT^{2} \) |
| 13 | \( 1 + (-10.4 + 10.4i)T - 169iT^{2} \) |
| 17 | \( 1 + 10.0iT - 289T^{2} \) |
| 19 | \( 1 + (-5.07 + 5.07i)T - 361iT^{2} \) |
| 23 | \( 1 + 29.7T + 529T^{2} \) |
| 29 | \( 1 + (-18.6 - 18.6i)T + 841iT^{2} \) |
| 31 | \( 1 + 27.1T + 961T^{2} \) |
| 37 | \( 1 + (13.3 + 13.3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 34.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-7.29 - 7.29i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 51.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (68.4 - 68.4i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (31.5 - 31.5i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (72.6 - 72.6i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-60.3 + 60.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 3.09T + 5.04e3T^{2} \) |
| 73 | \( 1 - 2.05iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 53.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (21.7 + 21.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 137.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 17.1T + 9.40e3T^{2} \) |
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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
−9.321739873694293049792413811931, −8.068796086652351049673685408888, −7.46403760655215846920629040300, −6.99435216410344462680148762823, −5.79637650018687453890583543804, −4.69199591335739481339781972569, −3.86750564711674352300532844168, −3.01309694431867534548258508628, −1.37830804296060379907881967617, −0.11993443027618675438452955714,
1.74696040743076478804693981287, 2.75005112445953073405816449891, 3.88529491056880555474615250738, 5.00358959247122946073233622625, 5.94808396862598779373551249900, 6.31873584107749465388023731589, 7.996118948605021909956688176321, 8.317217472294481222278574827877, 9.081402000871583412019286052546, 9.861761127055953932547115178371
