L(s) = 1 | + (−0.233 − 0.169i)2-s + (0.309 − 0.951i)3-s + (−0.592 − 1.82i)4-s + (−0.233 + 0.169i)6-s + (1.13 + 3.48i)7-s + (−0.349 + 1.07i)8-s + (−0.809 − 0.587i)9-s + (1.25 + 3.06i)11-s − 1.91·12-s + (3.90 + 2.83i)13-s + (0.327 − 1.00i)14-s + (−2.83 + 2.06i)16-s + (3.10 − 2.25i)17-s + (0.0892 + 0.274i)18-s + (−2.15 + 6.62i)19-s + ⋯ |
L(s) = 1 | + (−0.165 − 0.120i)2-s + (0.178 − 0.549i)3-s + (−0.296 − 0.911i)4-s + (−0.0953 + 0.0693i)6-s + (0.428 + 1.31i)7-s + (−0.123 + 0.380i)8-s + (−0.269 − 0.195i)9-s + (0.378 + 0.925i)11-s − 0.553·12-s + (1.08 + 0.786i)13-s + (0.0874 − 0.269i)14-s + (−0.709 + 0.515i)16-s + (0.753 − 0.547i)17-s + (0.0210 + 0.0647i)18-s + (−0.493 + 1.51i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45627 + 0.0716134i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45627 + 0.0716134i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (-1.25 - 3.06i)T \) |
good | 2 | \( 1 + (0.233 + 0.169i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + (-1.13 - 3.48i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.90 - 2.83i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.10 + 2.25i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.15 - 6.62i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 + (-1.52 - 4.70i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (5.96 + 4.33i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.737 - 2.27i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.33 + 4.09i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 0.772T + 43T^{2} \) |
| 47 | \( 1 + (-1.99 + 6.15i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-8.70 - 6.32i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (4.48 + 13.7i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.48 + 1.07i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 + (-2.66 + 1.93i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.0591 + 0.182i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.55 - 1.85i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.23 + 0.899i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 3.04T + 89T^{2} \) |
| 97 | \( 1 + (-11.1 - 8.09i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.14463810920788897817073682917, −9.192814397906002032889570099134, −8.798455573836575315514546568282, −7.81253900691015778858792946539, −6.63129403581935427186217809796, −5.84978318542435395771121795469, −5.11434736484242651604482983122, −3.82813603258602494331740341673, −2.16863939132976839049761421145, −1.47151076826667409273287509821,
0.845260140361262917017307183508, 3.03645110813382981046801313946, 3.80142260717254730568013824220, 4.52040699778426820142118298112, 5.81547284613358430609924070189, 6.97033197615254948254293861877, 7.82032768556072433656362678785, 8.522815926333548366156429772829, 9.145544721751620366978884627092, 10.34042345427924169796503325020