| L(s) = 1 | − 2i·3-s + 2i·7-s − 9-s + 2i·13-s − 6i·17-s − 4·19-s + 4·21-s − 6i·23-s − 4i·27-s + 6·29-s − 4·31-s − 2i·37-s + 4·39-s + 6·41-s − 10i·43-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + 0.755i·7-s − 0.333·9-s + 0.554i·13-s − 1.45i·17-s − 0.917·19-s + 0.872·21-s − 1.25i·23-s − 0.769i·27-s + 1.11·29-s − 0.718·31-s − 0.328i·37-s + 0.640·39-s + 0.937·41-s − 1.52i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.438308360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.438308360\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.864181538070128002851988081823, −8.465104758803893787805325964614, −7.34619343535064306042996878642, −6.85224633743913090682177552781, −6.09434938433328747182718098873, −5.12032888678831571270740769770, −4.12878098925751391315630894068, −2.63561954949853613425876041889, −2.05174102390501109448796544699, −0.58128801619554062481232200609,
1.38482072697619602370463073392, 2.98265302954566036668057108245, 3.98231194533867730950445797965, 4.39855579451985909396387400672, 5.49539059958553844134401934777, 6.30625831673245170586613956228, 7.37457652619397995200233679116, 8.148814130656262106238896035439, 9.003077847639080326930611714784, 9.796311374032066566307355733499
