| L(s) = 1 | + (0.178 − 1.69i)2-s + (1.53 + 1.70i)3-s + (−0.894 − 0.190i)4-s + (3.71 + 1.65i)5-s + (3.16 − 2.29i)6-s + (0.572 − 1.76i)8-s + (−0.235 + 2.23i)9-s + (3.47 − 6.01i)10-s + (−2.53 − 2.13i)11-s + (−1.04 − 1.81i)12-s + (−2.65 − 1.92i)13-s + (2.87 + 8.86i)15-s + (−4.56 − 2.03i)16-s + (0.138 + 1.31i)17-s + (3.75 + 0.799i)18-s + (−2.11 + 0.449i)19-s + ⋯ |
| L(s) = 1 | + (0.126 − 1.20i)2-s + (0.885 + 0.983i)3-s + (−0.447 − 0.0950i)4-s + (1.66 + 0.739i)5-s + (1.29 − 0.938i)6-s + (0.202 − 0.623i)8-s + (−0.0784 + 0.746i)9-s + (1.09 − 1.90i)10-s + (−0.764 − 0.644i)11-s + (−0.302 − 0.523i)12-s + (−0.735 − 0.534i)13-s + (0.743 + 2.28i)15-s + (−1.14 − 0.507i)16-s + (0.0334 + 0.318i)17-s + (0.886 + 0.188i)18-s + (−0.485 + 0.103i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.51073 - 0.707697i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.51073 - 0.707697i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + (2.53 + 2.13i)T \) |
| good | 2 | \( 1 + (-0.178 + 1.69i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (-1.53 - 1.70i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (-3.71 - 1.65i)T + (3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (2.65 + 1.92i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.138 - 1.31i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (2.11 - 0.449i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.933 - 1.61i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0754 + 0.232i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.27 - 2.79i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.171 + 0.189i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (1.77 - 5.45i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 + (-3.97 + 0.845i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (4.57 - 2.03i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-0.962 - 0.204i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-1.68 - 0.751i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (-1.50 + 2.60i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.23 - 3.80i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-9.44 - 2.00i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-0.578 + 5.50i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (1.67 - 1.21i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (8.31 + 14.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.09 - 1.51i)T + (29.9 + 92.2i)T^{2} \) |
| 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
−10.52544109499467402150121564892, −10.01245501608666253581452604093, −9.498786659142732117699814111864, −8.520622090805756594064993309262, −7.12138543090425909669488911722, −5.92612326896642068573387207307, −4.85490067336097120846809846165, −3.40152297269364744343795330240, −2.84450233935936455303112552704, −1.91645992696309045661738159106,
1.88709535490715636185217382635, 2.40599694217479839503402383531, 4.80109065475859835109939574876, 5.46393871024833460121241672172, 6.55247575915179563893006774882, 7.17868290191724486198501474111, 8.078206124316179498247244692708, 8.890034878270776831054696642362, 9.605937892452404449094796740078, 10.66835296551937936740258960531
