L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 11.5i·5-s + 22.6i·8-s + 32.6·10-s + 212. i·11-s + 288.·13-s + 64.0·16-s − 397. i·17-s − 346.·19-s − 92.2i·20-s + 601.·22-s − 659. i·23-s + 491.·25-s − 815. i·26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.461i·5-s + 0.353i·8-s + 0.326·10-s + 1.75i·11-s + 1.70·13-s + 0.250·16-s − 1.37i·17-s − 0.960·19-s − 0.230i·20-s + 1.24·22-s − 1.24i·23-s + 0.787·25-s − 1.20i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.863687588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.863687588\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 11.5iT - 625T^{2} \) |
| 11 | \( 1 - 212. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 288.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 397. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 346.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 659. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 192. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 613.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.17e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.35e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.25e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.12e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 112. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 978. iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 350.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 24.5T + 2.01e7T^{2} \) |
| 71 | \( 1 - 162. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.45e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.21e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 3.47e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.75e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.50e3T + 8.85e7T^{2} \) |
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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
−9.754698087518179367250099059322, −8.968484086833964282885024384604, −8.119208009695231096224741044915, −6.98129687744226179870553365339, −6.34914362404427169200407619789, −4.92542031283164662626280074675, −4.26704447425099172775542841195, −3.08217609240222035998315557663, −2.15898030769220523766273921371, −1.00607905567469007805570307356,
0.49678324598770265075595865843, 1.55381401603853506552119747040, 3.42590062080290421043576420073, 3.98905131689427652285168661946, 5.42001223904793601971619270735, 5.97846731239746181604494400103, 6.71827786514057450677953568655, 8.066283295146628855774930282564, 8.623440357900246360143973683170, 8.968565778098013490196724403099