L(s) = 1 | + (−0.498 − 1.32i)2-s + (−1.50 + 1.31i)4-s + (−2.06 − 1.19i)5-s + (0.866 − 0.5i)7-s + (2.49 + 1.33i)8-s + (−0.549 + 3.33i)10-s + (3.21 + 5.57i)11-s + (2.15 − 3.72i)13-s + (−1.09 − 0.896i)14-s + (0.516 − 3.96i)16-s − 1.89i·17-s + 0.529i·19-s + (4.68 − 0.935i)20-s + (5.77 − 7.03i)22-s + (2.64 − 4.58i)23-s + ⋯ |
L(s) = 1 | + (−0.352 − 0.935i)2-s + (−0.751 + 0.659i)4-s + (−0.925 − 0.534i)5-s + (0.327 − 0.188i)7-s + (0.882 + 0.470i)8-s + (−0.173 + 1.05i)10-s + (0.970 + 1.68i)11-s + (0.596 − 1.03i)13-s + (−0.292 − 0.239i)14-s + (0.129 − 0.991i)16-s − 0.459i·17-s + 0.121i·19-s + (1.04 − 0.209i)20-s + (1.23 − 1.50i)22-s + (0.552 − 0.956i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.561469 - 0.845739i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.561469 - 0.845739i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.498 + 1.32i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
good | 5 | \( 1 + (2.06 + 1.19i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.21 - 5.57i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.15 + 3.72i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.89iT - 17T^{2} \) |
| 19 | \( 1 - 0.529iT - 19T^{2} \) |
| 23 | \( 1 + (-2.64 + 4.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.301 + 0.174i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (5.09 + 2.94i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 0.842T + 37T^{2} \) |
| 41 | \( 1 + (4.58 + 2.64i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.38 + 4.26i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.04 - 3.53i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12.5iT - 53T^{2} \) |
| 59 | \( 1 + (-2.66 + 4.61i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.49 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.32 - 5.38i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 6.12T + 71T^{2} \) |
| 73 | \( 1 - 5.07T + 73T^{2} \) |
| 79 | \( 1 + (-2.87 + 1.66i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.23 + 5.60i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14.4iT - 89T^{2} \) |
| 97 | \( 1 + (3.86 + 6.69i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.09802040821017203482328517183, −9.305273749327673329928768133757, −8.455851591674769000104872956994, −7.74167646148299214064666759353, −6.89797043897862152607741006866, −5.16513597130658022789481167107, −4.33036003225327699847484843191, −3.61647890571621994548407716351, −2.09539139788206258392989750367, −0.71597099816949439901423762315,
1.25206181704047402618124364237, 3.46317494470370882069613049472, 4.16495363306963121633690424312, 5.54053826920737823257332390839, 6.35546617820043487233718900476, 7.14247096172472239389963096092, 8.034295500979200004370100104465, 8.821787199244529707257701450139, 9.324776714051510642454166495543, 10.84637877738333285092114374244