| L(s) = 1 | + 1.47i·2-s + 1.82·4-s + 4.22·5-s − 2.20i·7-s + 8.59i·8-s + 6.24i·10-s + 21.4i·13-s + 3.25·14-s − 5.40·16-s + 10.1i·17-s + 3.99i·19-s + 7.69·20-s − 0.0490·23-s − 7.12·25-s − 31.6·26-s + ⋯ |
| L(s) = 1 | + 0.738i·2-s + 0.455·4-s + 0.845·5-s − 0.314i·7-s + 1.07i·8-s + 0.624i·10-s + 1.65i·13-s + 0.232·14-s − 0.337·16-s + 0.595i·17-s + 0.210i·19-s + 0.384·20-s − 0.00213·23-s − 0.284·25-s − 1.21·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.498365277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.498365277\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.47iT - 4T^{2} \) |
| 5 | \( 1 - 4.22T + 25T^{2} \) |
| 7 | \( 1 + 2.20iT - 49T^{2} \) |
| 13 | \( 1 - 21.4iT - 169T^{2} \) |
| 17 | \( 1 - 10.1iT - 289T^{2} \) |
| 19 | \( 1 - 3.99iT - 361T^{2} \) |
| 23 | \( 1 + 0.0490T + 529T^{2} \) |
| 29 | \( 1 + 30.6iT - 841T^{2} \) |
| 31 | \( 1 + 25.0T + 961T^{2} \) |
| 37 | \( 1 - 4.51T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 68.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 64.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 12.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 49.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 91.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 22.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 69.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 35.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 130. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 128.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 86.8T + 9.40e3T^{2} \) |
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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
−9.769067102209296869829805576975, −9.164549316145032084343417740255, −8.114396969470320759859693896877, −7.37993668569303485322937034772, −6.37953196343741031081080845998, −6.10676805526068722309965898357, −4.96368154326265554187322975918, −3.91794227194214796824833050795, −2.39804397122551980277516147776, −1.59845232836419945504898286199,
0.71747499086990594245825260542, 2.00669812513903622968343169957, 2.80953092656535001235042371962, 3.73920379629031765792075898141, 5.29607098991708982027352873169, 5.80199928061661033066234037207, 6.91282334218124618807481442525, 7.65153526954950332947922281875, 8.842940535968525765297449758001, 9.560525716279062867554089702886
