L(s) = 1 | + (0.262 − 2.98i)3-s + (−1.10 + 1.90i)5-s + (−7.23 + 4.17i)7-s + (−8.86 − 1.56i)9-s + (−4.54 + 2.62i)11-s + (7.37 − 12.7i)13-s + (5.40 + 3.79i)15-s + 28.2·17-s + 19.1i·19-s + (10.5 + 22.7i)21-s + (−3.16 − 1.82i)23-s + (10.0 + 17.4i)25-s + (−7.00 + 26.0i)27-s + (12.3 + 21.3i)29-s + (32.9 + 19.0i)31-s + ⋯ |
L(s) = 1 | + (0.0874 − 0.996i)3-s + (−0.220 + 0.381i)5-s + (−1.03 + 0.597i)7-s + (−0.984 − 0.174i)9-s + (−0.413 + 0.238i)11-s + (0.567 − 0.982i)13-s + (0.360 + 0.252i)15-s + 1.66·17-s + 1.00i·19-s + (0.504 + 1.08i)21-s + (−0.137 − 0.0794i)23-s + (0.403 + 0.698i)25-s + (−0.259 + 0.965i)27-s + (0.425 + 0.736i)29-s + (1.06 + 0.613i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.294753988\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.294753988\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.262 + 2.98i)T \) |
good | 5 | \( 1 + (1.10 - 1.90i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (7.23 - 4.17i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (4.54 - 2.62i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-7.37 + 12.7i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 28.2T + 289T^{2} \) |
| 19 | \( 1 - 19.1iT - 361T^{2} \) |
| 23 | \( 1 + (3.16 + 1.82i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-12.3 - 21.3i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-32.9 - 19.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 4.21T + 1.36e3T^{2} \) |
| 41 | \( 1 + (9.92 - 17.1i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-20.1 + 11.6i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (25.8 - 14.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (7.96 + 4.59i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-40.8 - 70.7i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-6.86 - 3.96i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 62.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (53.7 - 31.0i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (103. - 59.4i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 107.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-1.78 - 3.09i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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
−10.50677531066828747022569412517, −9.814840296176898537095206838755, −8.592987930979136284608560808497, −7.901418612617048332843820634947, −7.02708532383840763649665100840, −6.05255059168077623358648124413, −5.42400519103211018047303273120, −3.39020293306135367878578921000, −2.84831356543961314235016088756, −1.15353387266312445358071011527,
0.57313952389063208341393508168, 2.81588031343660404846401460982, 3.79112167312614413279292302333, 4.63201112998586215931207813110, 5.78383419276047289826931575105, 6.73123746760304110026524535953, 7.983122836410068373365952077429, 8.797771939536551349162564400213, 9.780362185783525762414872109222, 10.16745703288228521312829219730