| L(s) = 1 | + (−0.0875 − 0.0234i)2-s + (2.77 + 1.13i)3-s + (−3.45 − 1.99i)4-s + (4.80 − 1.37i)5-s + (−0.216 − 0.164i)6-s + (2.18 − 6.65i)7-s + (0.512 + 0.512i)8-s + (6.41 + 6.31i)9-s + (−0.453 + 0.00753i)10-s + (6.21 + 3.58i)11-s + (−7.32 − 9.47i)12-s + (2.42 + 2.42i)13-s + (−0.346 + 0.531i)14-s + (14.9 + 1.65i)15-s + (7.95 + 13.7i)16-s + (−17.1 + 4.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.0437 − 0.0117i)2-s + (0.925 + 0.379i)3-s + (−0.864 − 0.498i)4-s + (0.961 − 0.274i)5-s + (−0.0360 − 0.0274i)6-s + (0.311 − 0.950i)7-s + (0.0640 + 0.0640i)8-s + (0.712 + 0.701i)9-s + (−0.0453 + 0.000753i)10-s + (0.565 + 0.326i)11-s + (−0.610 − 0.789i)12-s + (0.186 + 0.186i)13-s + (−0.0247 + 0.0379i)14-s + (0.993 + 0.110i)15-s + (0.496 + 0.860i)16-s + (−1.01 + 0.270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.70201 - 0.231461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70201 - 0.231461i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.77 - 1.13i)T \) |
| 5 | \( 1 + (-4.80 + 1.37i)T \) |
| 7 | \( 1 + (-2.18 + 6.65i)T \) |
| good | 2 | \( 1 + (0.0875 + 0.0234i)T + (3.46 + 2i)T^{2} \) |
| 11 | \( 1 + (-6.21 - 3.58i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-2.42 - 2.42i)T + 169iT^{2} \) |
| 17 | \( 1 + (17.1 - 4.60i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (6.28 + 10.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (0.0951 - 0.355i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 51.7T + 841T^{2} \) |
| 31 | \( 1 + (-2.02 - 1.16i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (4.65 - 17.3i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 44.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (45.6 - 45.6i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (14.6 - 54.7i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (-63.8 + 17.1i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-65.6 - 37.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-93.3 + 53.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (29.9 - 8.01i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 69.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-26.1 + 7.00i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-68.4 + 39.5i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (77.2 - 77.2i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-18.4 + 10.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (45.5 - 45.5i)T - 9.40e3iT^{2} \) |
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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
−13.39685898272473615928908481726, −13.18381133732684940490078403026, −10.98706377620634401386545313838, −9.981018322234444897132605491148, −9.259291500638060457367513254649, −8.351997024845137612678622174910, −6.73256341436979222419471115661, −4.99235045873402005889981082942, −4.00162419239024224701285045133, −1.69843231552669616940754549632,
2.11652099762157364958750364124, 3.68672503049185785509317888534, 5.47125816509984864068410708864, 6.94134543153785729880670181212, 8.479869251156742358684679327204, 8.968415575798710157549654451087, 9.977560228605197857998599347141, 11.69367406616808279713381282270, 12.91330869036835424092657125837, 13.50282604319791319131872913472
