| L(s) = 1 | + (−0.215 + 1.98i)2-s + (0.455 + 0.788i)3-s + (−3.90 − 0.856i)4-s + (−3.17 + 5.49i)5-s + (−1.66 + 0.735i)6-s + (3.79 + 5.88i)7-s + (2.54 − 7.58i)8-s + (4.08 − 7.07i)9-s + (−10.2 − 7.49i)10-s + (−11.4 + 6.60i)11-s + (−1.10 − 3.47i)12-s + 19.4·13-s + (−12.5 + 6.27i)14-s − 5.77·15-s + (14.5 + 6.69i)16-s + (13.7 − 7.96i)17-s + ⋯ |
| L(s) = 1 | + (−0.107 + 0.994i)2-s + (0.151 + 0.262i)3-s + (−0.976 − 0.214i)4-s + (−0.634 + 1.09i)5-s + (−0.277 + 0.122i)6-s + (0.541 + 0.840i)7-s + (0.318 − 0.948i)8-s + (0.453 − 0.786i)9-s + (−1.02 − 0.749i)10-s + (−1.04 + 0.600i)11-s + (−0.0919 − 0.289i)12-s + 1.49·13-s + (−0.894 + 0.447i)14-s − 0.385·15-s + (0.908 + 0.418i)16-s + (0.811 − 0.468i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.451733 + 0.905602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451733 + 0.905602i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.215 - 1.98i)T \) |
| 7 | \( 1 + (-3.79 - 5.88i)T \) |
| good | 3 | \( 1 + (-0.455 - 0.788i)T + (-4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (3.17 - 5.49i)T + (-12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (11.4 - 6.60i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 19.4T + 169T^{2} \) |
| 17 | \( 1 + (-13.7 + 7.96i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-8.22 + 14.2i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (11.9 - 20.7i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 16.6iT - 841T^{2} \) |
| 31 | \( 1 + (11.1 - 6.42i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (41.1 + 23.7i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 6.49iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 33.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (18.9 + 10.9i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-32.2 + 18.5i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-27.3 - 47.3i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-5.12 + 8.87i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-14.8 + 8.56i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 32.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-92.8 + 53.5i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-29.1 + 50.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 36.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-0.929 - 0.536i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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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
−15.52242244120066472739851634708, −14.73090816792929805534355918773, −13.54010230293605133177857406720, −12.02998308724928900709961307316, −10.64144360613222108406069248182, −9.330411507420773609726173524360, −8.014901488097978463759040191889, −6.94498156987843674121800300598, −5.44026911518550090821274839690, −3.60311024262593930223912054994,
1.23524191565847286561529707925, 3.81826012161585504401087623629, 5.14951671565236989462851640618, 8.026322386165996664899632740773, 8.336401450374709200456880015766, 10.27108279407152552362000308633, 11.09511848097719180778308671460, 12.45067298144958045979008107356, 13.27470155736959698130044537947, 14.11040427708172229612843994825
