L(s) = 1 | + 5.38i·2-s − 20.9·4-s − 18.3·5-s − 69.8i·8-s − 98.6i·10-s − 23.7i·11-s + 11.7i·13-s + 208.·16-s − 96.4·17-s − 21.0i·19-s + 384.·20-s + 127.·22-s + 152. i·23-s + 210.·25-s − 63.5·26-s + ⋯ |
L(s) = 1 | + 1.90i·2-s − 2.62·4-s − 1.63·5-s − 3.08i·8-s − 3.11i·10-s − 0.651i·11-s + 0.251i·13-s + 3.25·16-s − 1.37·17-s − 0.253i·19-s + 4.29·20-s + 1.23·22-s + 1.38i·23-s + 1.68·25-s − 0.478·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6209521678\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6209521678\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 5.38iT - 8T^{2} \) |
| 5 | \( 1 + 18.3T + 125T^{2} \) |
| 11 | \( 1 + 23.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 11.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 96.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 21.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 152. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 77.5iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 199. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 164.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 435.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 106.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 11.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 110. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 829.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 22.5iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 398.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 228. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 266. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 920.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 642.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 445.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.61e3iT - 9.12e5T^{2} \) |
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
−11.06855049757602210795749817024, −9.553922438563175739551247535308, −8.643152700106831687676612859912, −8.082297668281181785282024511703, −7.23771537747491060502972885374, −6.56526632051916678764608664046, −5.35109790678227115248149633027, −4.35875705373978462314574254062, −3.58533521044323061200118233737, −0.41094310701471828200919706622,
0.58984062420914842167840678731, 2.19354757854843679936273371216, 3.35896982062368105653203022524, 4.25874228216624919670704777010, 4.87753746869044404220140051652, 6.89774518080800707195040083069, 8.240307803014562175495569854380, 8.684250702566612185401629379288, 9.902208555806179851801325987929, 10.70630151995561929444023495192