| L(s) = 1 | + 1.41i·2-s + (−1.28 + 2.70i)3-s − 2.00·4-s + (−3.83 − 1.82i)6-s − 2.64·7-s − 2.82i·8-s + (−5.68 − 6.97i)9-s + 10.4i·11-s + (2.57 − 5.41i)12-s − 11.9·13-s − 3.74i·14-s + 4.00·16-s + 29.1i·17-s + (9.86 − 8.03i)18-s − 20.3·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.429 + 0.903i)3-s − 0.500·4-s + (−0.638 − 0.303i)6-s − 0.377·7-s − 0.353i·8-s + (−0.631 − 0.775i)9-s + 0.951i·11-s + (0.214 − 0.451i)12-s − 0.922·13-s − 0.267i·14-s + 0.250·16-s + 1.71i·17-s + (0.548 − 0.446i)18-s − 1.06·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2702306172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2702306172\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (1.28 - 2.70i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 11 | \( 1 - 10.4iT - 121T^{2} \) |
| 13 | \( 1 + 11.9T + 169T^{2} \) |
| 17 | \( 1 - 29.1iT - 289T^{2} \) |
| 19 | \( 1 + 20.3T + 361T^{2} \) |
| 23 | \( 1 - 7.71iT - 529T^{2} \) |
| 29 | \( 1 + 47.4iT - 841T^{2} \) |
| 31 | \( 1 + 35.5T + 961T^{2} \) |
| 37 | \( 1 - 58.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 85.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 42.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 37.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 53.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 58.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 67.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 19.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 33.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 46.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 65.1T + 9.40e3T^{2} \) |
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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
−9.720492864656283809130113004973, −8.917502623990821521525824631763, −8.001634404494838783582619878456, −7.03110840448094796417931874883, −6.14300568630594672412516945239, −5.46969064789441648068950893691, −4.33944064183174032568837753250, −3.88411403527606105438540858673, −2.24128685907349743669718115554, −0.10785393399637922204393278297,
0.948246201550784820682208788663, 2.38291706762737912191116344069, 3.11307240096005354808509118559, 4.60942797673481976416942791207, 5.44167621399588215892200125009, 6.42869387714491144582146866288, 7.24904069149520738531299269205, 8.125417195530473705094500057544, 9.054724056516118163002412672045, 9.785496201425590790800520951037
