| L(s) = 1 | + (−2.38 + 1.82i)3-s + 5.89i·5-s + (−0.351 + 0.608i)7-s + (2.33 − 8.69i)9-s + (15.3 + 8.85i)11-s + (7.81 − 13.5i)13-s + (−10.7 − 14.0i)15-s + (19.3 + 11.1i)17-s + (18.0 − 5.79i)19-s + (−0.275 − 2.09i)21-s + (2.81 + 1.62i)23-s − 9.73·25-s + (10.3 + 24.9i)27-s + 7.47i·29-s + (4.28 + 7.41i)31-s + ⋯ |
| L(s) = 1 | + (−0.793 + 0.608i)3-s + 1.17i·5-s + (−0.0502 + 0.0869i)7-s + (0.259 − 0.965i)9-s + (1.39 + 0.805i)11-s + (0.601 − 1.04i)13-s + (−0.717 − 0.935i)15-s + (1.14 + 0.658i)17-s + (0.952 − 0.305i)19-s + (−0.0130 − 0.0995i)21-s + (0.122 + 0.0705i)23-s − 0.389·25-s + (0.382 + 0.924i)27-s + 0.257i·29-s + (0.138 + 0.239i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0862 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0862 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.599828561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.599828561\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.38 - 1.82i)T \) |
| 19 | \( 1 + (-18.0 + 5.79i)T \) |
| good | 5 | \( 1 - 5.89iT - 25T^{2} \) |
| 7 | \( 1 + (0.351 - 0.608i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-15.3 - 8.85i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-7.81 + 13.5i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-19.3 - 11.1i)T + (144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-2.81 - 1.62i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 7.47iT - 841T^{2} \) |
| 31 | \( 1 + (-4.28 - 7.41i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 37.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (3.07 + 5.32i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 - 71.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (20.4 - 11.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 54.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 69.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-51.2 + 88.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (7.22 + 4.17i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (39.6 - 68.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-58.3 - 101. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-86.3 - 49.8i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-4.82 + 2.78i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (52.8 + 91.5i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.59539289053094438384140275220, −9.829285155989826817129121776216, −9.073078628776858911785866396978, −7.68234692548280911544043199636, −6.80375164764158246646871559193, −6.08780404229705757089142076526, −5.16460944497679315828561507183, −3.83572266819483860362260972818, −3.16512446889873580961046599573, −1.22877490829954410350645575314,
0.823709696096802832200616541568, 1.52969565876408834460198928214, 3.55617954880067975642355572680, 4.68902659542514595859065413704, 5.58370811685847973055450898236, 6.40297439679080343220455840446, 7.31733690518459551194445955478, 8.392997608003891570474043995397, 9.101019665141183631253128825823, 9.983128659104325630339601290235
