| L(s) = 1 | + (1.73 − 2i)7-s − 2·13-s + 8i·19-s − 5·25-s − 10.3·31-s + 6.92i·37-s − 10.3·43-s + (−1.00 − 6.92i)49-s + 14·61-s − 3.46·67-s + 13.8i·73-s − 4i·79-s + (−3.46 + 4i)91-s + 13.8i·97-s + 3.46·103-s + ⋯ |
| L(s) = 1 | + (0.654 − 0.755i)7-s − 0.554·13-s + 1.83i·19-s − 25-s − 1.86·31-s + 1.13i·37-s − 1.58·43-s + (−0.142 − 0.989i)49-s + 1.79·61-s − 0.423·67-s + 1.62i·73-s − 0.450i·79-s + (−0.363 + 0.419i)91-s + 1.40i·97-s + 0.341·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7596064660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7596064660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.479537497802696202629711006385, −7.952568936946367042604115806505, −7.34492989152120649527121991782, −6.57525485709150023010759836653, −5.61955653724020712870269442155, −5.01860074071379287355490215441, −4.00330661170997031533123748273, −3.51153567569268180409215748451, −2.12642650724445793238324389250, −1.35350069480209126611802736575,
0.20611579941423764907065112433, 1.79486262186254845523773444626, 2.47730890764769768314090476273, 3.51296938052169774684316213301, 4.54209176860720060275450311583, 5.21432803452624977476462048041, 5.81170964435799979733276339506, 6.86554798840573066971062532355, 7.42090258634286986583662906347, 8.229465644192799591011220152078
