| L(s) = 1 | + 2.79i·3-s − 0.208i·7-s − 4.79·9-s − 11-s + i·13-s + 0.791i·17-s − 6.58·19-s + 0.582·21-s + 3.79i·23-s − 4.99i·27-s − 6.79·29-s − 8.58·31-s − 2.79i·33-s − 2.58i·37-s − 2.79·39-s + ⋯ |
| L(s) = 1 | + 1.61i·3-s − 0.0788i·7-s − 1.59·9-s − 0.301·11-s + 0.277i·13-s + 0.191i·17-s − 1.51·19-s + 0.127·21-s + 0.790i·23-s − 0.962i·27-s − 1.26·29-s − 1.54·31-s − 0.485i·33-s − 0.424i·37-s − 0.446·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7028774546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7028774546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.79iT - 3T^{2} \) |
| 7 | \( 1 + 0.208iT - 7T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 0.791iT - 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 - 3.79iT - 23T^{2} \) |
| 29 | \( 1 + 6.79T + 29T^{2} \) |
| 31 | \( 1 + 8.58T + 31T^{2} \) |
| 37 | \( 1 + 2.58iT - 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 1.41iT - 47T^{2} \) |
| 53 | \( 1 + 11.3iT - 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 4.20T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 7.79iT - 73T^{2} \) |
| 79 | \( 1 - 15.5T + 79T^{2} \) |
| 83 | \( 1 - 9.95iT - 83T^{2} \) |
| 89 | \( 1 - 0.791T + 89T^{2} \) |
| 97 | \( 1 + 6.20iT - 97T^{2} \) |
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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
−10.29013617444085019982658518598, −9.538303344521521368406262328182, −8.941654217194514018962726732689, −8.085789586865989827234387374665, −6.97071852554104483422356760541, −5.79907829755189068667780124479, −5.10945551890083596516048082373, −4.09564233582927254930854269768, −3.56177444625620851077238643058, −2.14914647901678809567186536410,
0.28892035120979027137618594892, 1.79352427427937454481491753910, 2.58724957592837577759758047796, 3.97658914887524779390489554965, 5.37280980341232620538171177257, 6.13394506985107170407321533750, 6.99816685704410290554520504568, 7.56909086594879375955636805197, 8.455218599543843519309855777610, 9.061381198071603676958120286937
