| L(s) = 1 | + (0.766 − 0.642i)3-s + (−2.43 − 0.884i)5-s + (0.439 − 0.761i)7-s + (0.173 − 0.984i)9-s + (−1.21 − 2.10i)11-s + (−0.646 − 0.542i)13-s + (−2.43 + 0.884i)15-s + (−0.715 − 4.05i)17-s + (1.96 − 3.89i)19-s + (−0.152 − 0.866i)21-s + (−6.42 + 2.33i)23-s + (1.29 + 1.08i)25-s + (−0.500 − 0.866i)27-s + (1.12 − 6.36i)29-s + (−1.77 + 3.07i)31-s + ⋯ |
| L(s) = 1 | + (0.442 − 0.371i)3-s + (−1.08 − 0.395i)5-s + (0.166 − 0.287i)7-s + (0.0578 − 0.328i)9-s + (−0.366 − 0.634i)11-s + (−0.179 − 0.150i)13-s + (−0.627 + 0.228i)15-s + (−0.173 − 0.983i)17-s + (0.451 − 0.892i)19-s + (−0.0333 − 0.188i)21-s + (−1.34 + 0.487i)23-s + (0.258 + 0.217i)25-s + (−0.0962 − 0.166i)27-s + (0.208 − 1.18i)29-s + (−0.318 + 0.552i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.561864 - 0.897669i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.561864 - 0.897669i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-1.96 + 3.89i)T \) |
| good | 5 | \( 1 + (2.43 + 0.884i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.439 + 0.761i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.21 + 2.10i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.646 + 0.542i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.715 + 4.05i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (6.42 - 2.33i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.12 + 6.36i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.77 - 3.07i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + (-4.11 + 3.45i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.41 + 0.514i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.90 - 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-4.74 + 1.72i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.601 - 3.40i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.48 + 0.539i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.662 + 3.75i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.0721 + 0.0262i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-9.29 + 7.80i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.51 + 1.27i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.23 + 10.7i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.84 + 2.38i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.82 - 16.0i)T + (-91.1 + 33.1i)T^{2} \) |
| 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
−11.00989825857059505034475505723, −9.743636285020505832652358706877, −8.855187237617181020267071634262, −7.80118730681607623363928563173, −7.54615256235852323474603959870, −6.15477614237762474541406345582, −4.83230993048813204256258136974, −3.83526817733118991111219274537, −2.61628170854422240487661161582, −0.62096301744160108040570930835,
2.16245064080096580926734234363, 3.58129911557840479557006447171, 4.32466042439574513701199654981, 5.61865183905237346665087641145, 6.93860931683969208029436618872, 7.911324285542953219713400228448, 8.407489964885849642470446875580, 9.684903993447692374736065542232, 10.40918922570282047169970197587, 11.35254009865978114237334829004
