| L(s) = 1 | + 15.1i·3-s − 12.2·5-s + 18.5i·7-s − 147.·9-s + 114. i·11-s − 7.25·13-s − 185. i·15-s − 194.·17-s − 699. i·19-s − 279.·21-s − 275. i·23-s − 474.·25-s − 1.00e3i·27-s + 1.03e3·29-s + 1.00e3i·31-s + ⋯ |
| L(s) = 1 | + 1.67i·3-s − 0.491·5-s + 0.377i·7-s − 1.81·9-s + 0.943i·11-s − 0.0429·13-s − 0.824i·15-s − 0.672·17-s − 1.93i·19-s − 0.634·21-s − 0.520i·23-s − 0.758·25-s − 1.37i·27-s + 1.22·29-s + 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.197581 - 0.737384i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.197581 - 0.737384i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 18.5iT \) |
| good | 3 | \( 1 - 15.1iT - 81T^{2} \) |
| 5 | \( 1 + 12.2T + 625T^{2} \) |
| 11 | \( 1 - 114. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 7.25T + 2.85e4T^{2} \) |
| 17 | \( 1 + 194.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 699. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 275. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.03e3T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.00e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 859.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 3.29e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.64e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.56e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.84e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.81e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.58e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.39e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 4.15e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 3.73e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.00e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.25e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.15e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 8.54e3T + 8.85e7T^{2} \) |
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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
−13.69577882645916660100817028883, −12.21848423105298742918047296635, −11.23718505170771922883091198212, −10.32073267581831853557480612470, −9.339490964614646091512300203402, −8.488135892705913063661259839855, −6.79400370845191835498694533192, −5.02532370082343705109314311691, −4.34235933213961729328753658617, −2.83507021488450752018177520203,
0.32649713108637932214704635783, 1.81773569821903738771649775000, 3.60203457657536376819953849403, 5.74024804869409988528927518566, 6.78934928882121409713021662302, 7.84657903561982362857791556954, 8.536830833268770700030627021619, 10.36214972208004095338870516038, 11.71191252879026363748051281819, 12.18201162627754581910000112223
