| L(s) = 1 | + 4.89i·5-s − 7.74i·7-s + 12.6·11-s + 15.4i·13-s + 25.2·17-s − 8·19-s + 39.1i·23-s + 1.00·25-s + 24.4i·29-s + 7.74i·31-s + 37.9·35-s − 46.4i·37-s + 25.2·41-s + 40·43-s − 39.1i·47-s + ⋯ |
| L(s) = 1 | + 0.979i·5-s − 1.10i·7-s + 1.14·11-s + 1.19i·13-s + 1.48·17-s − 0.421·19-s + 1.70i·23-s + 0.0400·25-s + 0.844i·29-s + 0.249i·31-s + 1.08·35-s − 1.25i·37-s + 0.617·41-s + 0.930·43-s − 0.833i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.61953 + 0.553878i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61953 + 0.553878i\) |
| \(L(2)\) |
|
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 | \( 1 - 4.89iT - 25T^{2} \) |
| 7 | \( 1 + 7.74iT - 49T^{2} \) |
| 11 | \( 1 - 12.6T + 121T^{2} \) |
| 13 | \( 1 - 15.4iT - 169T^{2} \) |
| 17 | \( 1 - 25.2T + 289T^{2} \) |
| 19 | \( 1 + 8T + 361T^{2} \) |
| 23 | \( 1 - 39.1iT - 529T^{2} \) |
| 29 | \( 1 - 24.4iT - 841T^{2} \) |
| 31 | \( 1 - 7.74iT - 961T^{2} \) |
| 37 | \( 1 + 46.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 25.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 39.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 25.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 15.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 80T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 + 54.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 139.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 50.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 50T + 9.40e3T^{2} \) |
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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
−11.52632982871255974755483712321, −10.80661083976295564300105231178, −9.880213926803905958049435570064, −8.998307383025532344923236603830, −7.41788989597509426312750939136, −7.02310083699353109952619861096, −5.85752672140085607083999079693, −4.19264559819605226257838137509, −3.35620786455081326585926285389, −1.45121295747060003886977510261,
1.02359550910949908901703368983, 2.78815526408337294670385706546, 4.34211128234409429557935947647, 5.48219054696703720462176692083, 6.29560691774140837787138103901, 7.908385297380946473028717944941, 8.642801727606034616623472558268, 9.444641270615219006078322888183, 10.47201359012448902455683490228, 11.81788301460158182956257735477
