| L(s) = 1 | + (−2.18 − 3.78i)2-s + (3.55 + 3.78i)3-s + (−5.55 + 9.62i)4-s + (−2.31 + 4.00i)5-s + (6.55 − 21.7i)6-s + (−6.05 − 10.4i)7-s + 13.6·8-s + (−1.67 + 26.9i)9-s + 20.2·10-s + (−5.01 − 8.67i)11-s + (−56.2 + 13.2i)12-s + (24.2 − 42.0i)13-s + (−26.4 + 45.8i)14-s + (−23.4 + 5.49i)15-s + (14.6 + 25.4i)16-s + 75.3·17-s + ⋯ |
| L(s) = 1 | + (−0.772 − 1.33i)2-s + (0.684 + 0.728i)3-s + (−0.694 + 1.20i)4-s + (−0.206 + 0.358i)5-s + (0.446 − 1.48i)6-s + (−0.327 − 0.566i)7-s + 0.602·8-s + (−0.0620 + 0.998i)9-s + 0.639·10-s + (−0.137 − 0.237i)11-s + (−1.35 + 0.317i)12-s + (0.518 − 0.897i)13-s + (−0.505 + 0.875i)14-s + (−0.402 + 0.0946i)15-s + (0.229 + 0.397i)16-s + 1.07·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.594 + 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.600462 - 0.303034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.600462 - 0.303034i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.55 - 3.78i)T \) |
| good | 2 | \( 1 + (2.18 + 3.78i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (2.31 - 4.00i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (6.05 + 10.4i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (5.01 + 8.67i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-24.2 + 42.0i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 75.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-19.0 + 32.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-11.3 - 19.5i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (15.0 - 26.0i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 130.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (173. - 300. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (13.3 + 23.1i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (230. + 399. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (4.18 - 7.24i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-41.0 - 71.0i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (341. - 591. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 1.09e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 + (243. + 420. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (49.5 + 85.8i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 8.80T + 7.04e5T^{2} \) |
| 97 | \( 1 + (330. + 572. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−20.61914174957277648201216313855, −19.61476470405738677535335294555, −18.59801363309162403456432729925, −16.77693799993086464251421774646, −14.95846146462468485433937781861, −13.03963571238007264459722957608, −10.95181254561589137292804741136, −10.02099796895845103617255852559, −8.338707171049150906418840569156, −3.34962024508304345866916773268,
6.42275905042016689314401269882, 8.085664835665595463502938803439, 9.256610890634567397347525655757, 12.45166949290232207508515373311, 14.31845763182110684557309037513, 15.58090525747337205004237575777, 16.94023726446081188549837194070, 18.42978494776154011052373199594, 19.23697264098579876652278687290, 21.04125960132502301779273003424
