L(s) = 1 | + (1.14 − 0.826i)2-s − 1.52i·3-s + (0.632 − 1.89i)4-s + (−0.0967 + 2.23i)5-s + (−1.25 − 1.74i)6-s + (−0.843 − 2.69i)8-s + 0.679·9-s + (1.73 + 2.64i)10-s − 4.56i·11-s + (−2.89 − 0.963i)12-s + 2.19·13-s + (3.40 + 0.147i)15-s + (−3.19 − 2.40i)16-s + 6.22·17-s + (0.779 − 0.561i)18-s − 3.83·19-s + ⋯ |
L(s) = 1 | + (0.811 − 0.584i)2-s − 0.879i·3-s + (0.316 − 0.948i)4-s + (−0.0432 + 0.999i)5-s + (−0.514 − 0.713i)6-s + (−0.298 − 0.954i)8-s + 0.226·9-s + (0.549 + 0.835i)10-s − 1.37i·11-s + (−0.834 − 0.278i)12-s + 0.608·13-s + (0.878 + 0.0380i)15-s + (−0.799 − 0.600i)16-s + 1.50·17-s + (0.183 − 0.132i)18-s − 0.879·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.29165 - 2.23386i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.29165 - 2.23386i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.14 + 0.826i)T \) |
| 5 | \( 1 + (0.0967 - 2.23i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 1.52iT - 3T^{2} \) |
| 11 | \( 1 + 4.56iT - 11T^{2} \) |
| 13 | \( 1 - 2.19T + 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 + 3.83T + 19T^{2} \) |
| 23 | \( 1 - 0.430T + 23T^{2} \) |
| 29 | \( 1 + 0.473T + 29T^{2} \) |
| 31 | \( 1 + 7.59T + 31T^{2} \) |
| 37 | \( 1 + 8.44iT - 37T^{2} \) |
| 41 | \( 1 - 1.45iT - 41T^{2} \) |
| 43 | \( 1 - 8.58T + 43T^{2} \) |
| 47 | \( 1 - 4.48iT - 47T^{2} \) |
| 53 | \( 1 - 9.23iT - 53T^{2} \) |
| 59 | \( 1 - 3.13T + 59T^{2} \) |
| 61 | \( 1 - 5.71iT - 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 - 4.57iT - 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 6.20iT - 79T^{2} \) |
| 83 | \( 1 + 7.69iT - 83T^{2} \) |
| 89 | \( 1 + 9.32iT - 89T^{2} \) |
| 97 | \( 1 - 9.05T + 97T^{2} \) |
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
−10.06570230359190322753586622785, −8.956815369911944628500543113706, −7.75494783676981323325458812452, −7.08048418991961835585649107361, −6.03351707005759433638055069831, −5.74892042551711505634782703979, −4.04311529465515283308754309382, −3.30598206463800243677566552165, −2.25553057471165263475756642871, −0.987496435801405717583412475037,
1.81463734633196023244345276950, 3.53161914072919417385298526437, 4.22643260115220278719577214147, 4.97816589470697343577471429736, 5.64366316478003762969033624459, 6.85498960355407723451963916785, 7.71821106436170566997033591296, 8.550217286851814963662354549569, 9.444927326350482244272864012419, 10.12883366770525072095377713443