| 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 | \( 1 + (-2.82 + 16.0i)T + (-91.1 - 33.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.35254009865978114237334829004, −10.40918922570282047169970197587, −9.684903993447692374736065542232, −8.407489964885849642470446875580, −7.911324285542953219713400228448, −6.93860931683969208029436618872, −5.61865183905237346665087641145, −4.32466042439574513701199654981, −3.58129911557840479557006447171, −2.16245064080096580926734234363,
0.62096301744160108040570930835, 2.61628170854422240487661161582, 3.83526817733118991111219274537, 4.83230993048813204256258136974, 6.15477614237762474541406345582, 7.54615256235852323474603959870, 7.80118730681607623363928563173, 8.855187237617181020267071634262, 9.743636285020505832652358706877, 11.00989825857059505034475505723
