| L(s) = 1 | + 2.43i·3-s − 5-s − 2.50·7-s − 2.90·9-s + (−0.620 + 3.25i)11-s − 3.68i·13-s − 2.43i·15-s − 1.43i·17-s − 4.51·19-s − 6.07i·21-s − 4.41i·23-s + 25-s + 0.227i·27-s + 1.75i·29-s − 2.47i·31-s + ⋯ |
| L(s) = 1 | + 1.40i·3-s − 0.447·5-s − 0.944·7-s − 0.968·9-s + (−0.187 + 0.982i)11-s − 1.02i·13-s − 0.627i·15-s − 0.347i·17-s − 1.03·19-s − 1.32i·21-s − 0.920i·23-s + 0.200·25-s + 0.0437i·27-s + 0.325i·29-s − 0.444i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4840097673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4840097673\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + (0.620 - 3.25i)T \) |
| good | 3 | \( 1 - 2.43iT - 3T^{2} \) |
| 7 | \( 1 + 2.50T + 7T^{2} \) |
| 13 | \( 1 + 3.68iT - 13T^{2} \) |
| 17 | \( 1 + 1.43iT - 17T^{2} \) |
| 19 | \( 1 + 4.51T + 19T^{2} \) |
| 23 | \( 1 + 4.41iT - 23T^{2} \) |
| 29 | \( 1 - 1.75iT - 29T^{2} \) |
| 31 | \( 1 + 2.47iT - 31T^{2} \) |
| 37 | \( 1 - 2.39T + 37T^{2} \) |
| 41 | \( 1 + 12.0iT - 41T^{2} \) |
| 43 | \( 1 - 3.44T + 43T^{2} \) |
| 47 | \( 1 - 2.65iT - 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 59 | \( 1 - 10.5iT - 59T^{2} \) |
| 61 | \( 1 + 10.1iT - 61T^{2} \) |
| 67 | \( 1 + 12.2iT - 67T^{2} \) |
| 71 | \( 1 + 8.04iT - 71T^{2} \) |
| 73 | \( 1 - 8.49iT - 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.332128136550178476498239070708, −8.613571079372672069192704278494, −7.64416642188000539843758163082, −6.77707586622226308204421891263, −5.79857439727252373109320956224, −4.86795166041232750274429244322, −4.19199314118999325845422046840, −3.40150880318863932974527278144, −2.45472072074783575627572704908, −0.19624949676250224919931227676,
1.16075830039958958833388319933, 2.35408198247466984742747651293, 3.35872054157250238565117206903, 4.33753243837184374026568726407, 5.74780951639890431315649446108, 6.43365273701928871459259340975, 6.91022590700306006979164162028, 7.83040442407798544249997946525, 8.440817208763615669599290019824, 9.245202571329600543220506552831
