L(s) = 1 | + (−1 − i)3-s + (−1.41 + 1.41i)5-s + 2.82i·7-s − i·9-s + (3 − 3i)11-s + (−4.24 − 4.24i)13-s + 2.82·15-s + (3 + 3i)19-s + (2.82 − 2.82i)21-s + 8.48i·23-s + 0.999i·25-s + (−4 + 4i)27-s + (1.41 + 1.41i)29-s + 5.65·31-s − 6·33-s + ⋯ |
L(s) = 1 | + (−0.577 − 0.577i)3-s + (−0.632 + 0.632i)5-s + 1.06i·7-s − 0.333i·9-s + (0.904 − 0.904i)11-s + (−1.17 − 1.17i)13-s + 0.730·15-s + (0.688 + 0.688i)19-s + (0.617 − 0.617i)21-s + 1.76i·23-s + 0.199i·25-s + (−0.769 + 0.769i)27-s + (0.262 + 0.262i)29-s + 1.01·31-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8673514019\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8673514019\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 5 | \( 1 + (1.41 - 1.41i)T - 5iT^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + (-3 + 3i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.24 + 4.24i)T + 13iT^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + (-3 - 3i)T + 19iT^{2} \) |
| 23 | \( 1 - 8.48iT - 23T^{2} \) |
| 29 | \( 1 + (-1.41 - 1.41i)T + 29iT^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (4.24 - 4.24i)T - 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (1.41 - 1.41i)T - 53iT^{2} \) |
| 59 | \( 1 + (1 - i)T - 59iT^{2} \) |
| 61 | \( 1 + (-4.24 - 4.24i)T + 61iT^{2} \) |
| 67 | \( 1 + (-9 - 9i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 + (3 + 3i)T + 83iT^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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
−10.04782725196154851674811644823, −9.336576107691955673425879426308, −8.284663865712695698311853242861, −7.53161215451432126812972597166, −6.70560504915944584189624283534, −5.83792865444966772395299209985, −5.23290258186996399404548335291, −3.57336545675302032102460671821, −2.92830861523978133311758417872, −1.21384734293322764981441273725,
0.48420951722120543289628319774, 2.17830728129228609676492663249, 4.00658069465890824964854811837, 4.51724982003431956238242800376, 5.01694568048205713192358262416, 6.63376834381554036946358940904, 7.13064758306052814034870161589, 8.099930836136042593427402510072, 9.117810221266076677761957087555, 9.899591242853628464837443665601