| L(s) = 1 | + (0.530 − 0.919i)2-s + (1.89 − 2.32i)3-s + (1.43 + 2.48i)4-s + (3.80 − 3.23i)5-s + (−1.12 − 2.97i)6-s + (−3.04 + 6.30i)7-s + 7.29·8-s + (−1.78 − 8.82i)9-s + (−0.956 − 5.22i)10-s + (−9.99 + 5.76i)11-s + (8.50 + 1.38i)12-s − 5.56i·13-s + (4.17 + 6.14i)14-s + (−0.290 − 14.9i)15-s + (−1.87 + 3.24i)16-s + (−7.82 − 13.5i)17-s + ⋯ |
| L(s) = 1 | + (0.265 − 0.459i)2-s + (0.633 − 0.774i)3-s + (0.359 + 0.622i)4-s + (0.761 − 0.647i)5-s + (−0.187 − 0.496i)6-s + (−0.435 + 0.900i)7-s + 0.912·8-s + (−0.198 − 0.980i)9-s + (−0.0956 − 0.522i)10-s + (−0.908 + 0.524i)11-s + (0.708 + 0.115i)12-s − 0.428i·13-s + (0.298 + 0.439i)14-s + (−0.0193 − 0.999i)15-s + (−0.117 + 0.202i)16-s + (−0.460 − 0.797i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.85334 - 0.885313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85334 - 0.885313i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.89 + 2.32i)T \) |
| 5 | \( 1 + (-3.80 + 3.23i)T \) |
| 7 | \( 1 + (3.04 - 6.30i)T \) |
| good | 2 | \( 1 + (-0.530 + 0.919i)T + (-2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (9.99 - 5.76i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 5.56iT - 169T^{2} \) |
| 17 | \( 1 + (7.82 + 13.5i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-1.06 + 1.84i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (13.0 - 22.5i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 13.3iT - 841T^{2} \) |
| 31 | \( 1 + (-25.3 - 43.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (19.8 + 11.4i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 53.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 38.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (2.72 - 4.71i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (15.5 + 26.8i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-97.9 + 56.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (14.8 - 25.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-23.5 + 13.6i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 79.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-1.70 + 0.981i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-43.8 + 75.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 71.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + (106. + 61.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 1.02iT - 9.40e3T^{2} \) |
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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
−13.12653478650076918130324539777, −12.55276599409250943508595487616, −11.74219736099284712466402143254, −10.07853361687672396767601649393, −8.935403867504325639425948300881, −7.935493895259607451763781282217, −6.68243821555956572934211585010, −5.16030903799092380079405817564, −3.06567165373098373285659821319, −2.01127520909562254602281258748,
2.42411786118666289230701219657, 4.17694861864766202407557029635, 5.68798096355925937048737711678, 6.76554116892735239336987602160, 8.080575220413170025047194022142, 9.705201900068705781317915807841, 10.39514138283385652267938396628, 11.02357280358997804789363196786, 13.30001749042499868209278809422, 13.82747724458106950609773236837
