| 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 | \( 1 + (-2.09 + 1.51i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.66835296551937936740258960531, −9.605937892452404449094796740078, −8.890034878270776831054696642362, −8.078206124316179498247244692708, −7.17868290191724486198501474111, −6.55247575915179563893006774882, −5.46393871024833460121241672172, −4.80109065475859835109939574876, −2.40599694217479839503402383531, −1.88709535490715636185217382635,
1.91645992696309045661738159106, 2.84450233935936455303112552704, 3.40152297269364744343795330240, 4.85490067336097120846809846165, 5.92612326896642068573387207307, 7.12138543090425909669488911722, 8.520622090805756594064993309262, 9.498786659142732117699814111864, 10.01245501608666253581452604093, 10.52544109499467402150121564892
