L(s) = 1 | − 11·7-s + (22.5 − 12.9i)13-s + (−13 + 13.8i)19-s + (12.5 + 21.6i)25-s + 60.6i·31-s + 57.1i·37-s + (−41.5 + 71.8i)43-s + 72·49-s + (−60.5 − 104. i)61-s + (115.5 − 66.6i)67-s + (−71.5 + 123. i)73-s + (76.5 + 44.1i)79-s + (−247.5 + 142. i)91-s + (168 + 96.9i)97-s + 133. i·103-s + ⋯ |
L(s) = 1 | − 1.57·7-s + (1.73 − 0.999i)13-s + (−0.684 + 0.729i)19-s + (0.5 + 0.866i)25-s + 1.95i·31-s + 1.54i·37-s + (−0.965 + 1.67i)43-s + 1.46·49-s + (−0.991 − 1.71i)61-s + (1.72 − 0.995i)67-s + (−0.979 + 1.69i)73-s + (0.968 + 0.559i)79-s + (−2.71 + 1.57i)91-s + (1.73 + 0.999i)97-s + 1.29i·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.125175280\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.125175280\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (13 - 13.8i)T \) |
good | 5 | \( 1 + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 11T + 49T^{2} \) |
| 11 | \( 1 + 121T^{2} \) |
| 13 | \( 1 + (-22.5 + 12.9i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 - 60.6iT - 961T^{2} \) |
| 37 | \( 1 - 57.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (41.5 - 71.8i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (60.5 + 104. i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-115.5 + 66.6i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (71.5 - 123. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-76.5 - 44.1i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 6.88e3T^{2} \) |
| 89 | \( 1 + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-168 - 96.9i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.43759936367809315199317138670, −9.692261486005965048110378838459, −8.729617651986676739183317895115, −8.023527167147012468724841163045, −6.60564286120665850318056002212, −6.28018464851400605090039095980, −5.12049721063222219240393062631, −3.57492036390584089042156084482, −3.13300107982041113196853556280, −1.22947968115215383432302855546,
0.44371192407274583291046544366, 2.24360501495517763523392124814, 3.53294852899361318296913311927, 4.27690279885097168009028848205, 5.90565837167600327482695798339, 6.40952927209120542109622744322, 7.25627531938221094505396265960, 8.666043772186825028148405398702, 9.094984965812344253487554075684, 10.07134004508359129576049360505