L(s) = 1 | + (−1.22 + 0.707i)2-s + (−2.36 + 1.84i)3-s + (0.999 − 1.73i)4-s + (1.59 − 3.93i)6-s + (4.79 + 8.30i)7-s + 2.82i·8-s + (2.18 − 8.73i)9-s + (8.83 − 5.10i)11-s + (0.833 + 5.94i)12-s + (1.72 − 2.99i)13-s + (−11.7 − 6.77i)14-s + (−2.00 − 3.46i)16-s − 30.5i·17-s + (3.50 + 12.2i)18-s − 10.0·19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.788 + 0.615i)3-s + (0.249 − 0.433i)4-s + (0.265 − 0.655i)6-s + (0.684 + 1.18i)7-s + 0.353i·8-s + (0.242 − 0.970i)9-s + (0.803 − 0.463i)11-s + (0.0694 + 0.495i)12-s + (0.132 − 0.230i)13-s + (−0.838 − 0.484i)14-s + (−0.125 − 0.216i)16-s − 1.79i·17-s + (0.194 + 0.679i)18-s − 0.530·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.105517353\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.105517353\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (2.36 - 1.84i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-4.79 - 8.30i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.83 + 5.10i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-1.72 + 2.99i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 30.5iT - 289T^{2} \) |
| 19 | \( 1 + 10.0T + 361T^{2} \) |
| 23 | \( 1 + (-26.5 - 15.3i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-36.3 + 21.0i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-5.01 + 8.69i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 21.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-34.3 - 19.8i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-10.8 - 18.7i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (10.8 - 6.28i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 47.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-28.0 - 16.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-57.1 - 98.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (16.1 - 27.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 65.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 54.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-44.0 - 76.3i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-35.5 + 20.5i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 38.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (64.1 + 111. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.20241823762783986909986380419, −9.986146759880054094488097511215, −9.152162537595736303875520729557, −8.603840045994664954531293159538, −7.27619240079683125214842876038, −6.23155852737279008500354326818, −5.42924820671149615928468459582, −4.54696326291378693540173530774, −2.81166156402013155358523486441, −0.948493022371463065269501081604,
0.935283626498210784715680001593, 1.87450757737040722224716677747, 3.88237658415755059785103253576, 4.84904589941211565962690105008, 6.44409499498247858614756091795, 6.97238759976344070114464970359, 8.011258066849484620462850000846, 8.795936940883402011469897535544, 10.29646400463692480410381382399, 10.66656048765304520644927302473