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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.969184703\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.969184703\) |
\(L(2)\) |
|
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 | \( 1 + (-1.78 - 3.09i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.73107123346896864508153248466, −9.877231848882238801351689231548, −8.841571514998992851704594191516, −8.018272230218097906587415562846, −7.12755943625442473024943495468, −5.72114602844958196810091566930, −5.01232757352160996184768015243, −3.85053672811319340858728730320, −3.05642643487255965698784868645, −1.00745278447312552539636702426,
1.15668047569459118796650827727, 2.06862169921385767463275898790, 3.68744105061778096697179385326, 5.02747651845718583754326175091, 5.87284218705774869222748359752, 6.88168074698410530218230447622, 7.952836566542583478635620253589, 8.408864912903222650436764388358, 9.355115476441920927388050889960, 10.62897572272293078572106360889