L(s) = 1 | + (−2.82 + 1.01i)3-s + (1.81 − 3.13i)5-s + (−1.59 + 0.920i)7-s + (6.93 − 5.73i)9-s + (10.0 − 5.80i)11-s + (6.43 − 11.1i)13-s + (−1.92 + 10.6i)15-s + 12.6·17-s − 25.6i·19-s + (3.56 − 4.21i)21-s + (25.9 + 14.9i)23-s + (5.93 + 10.2i)25-s + (−13.7 + 23.2i)27-s + (−10.8 − 18.7i)29-s + (−52.4 − 30.2i)31-s + ⋯ |
L(s) = 1 | + (−0.940 + 0.338i)3-s + (0.362 − 0.627i)5-s + (−0.227 + 0.131i)7-s + (0.770 − 0.637i)9-s + (0.914 − 0.528i)11-s + (0.495 − 0.857i)13-s + (−0.128 + 0.713i)15-s + 0.742·17-s − 1.35i·19-s + (0.169 − 0.200i)21-s + (1.12 + 0.650i)23-s + (0.237 + 0.411i)25-s + (−0.509 + 0.860i)27-s + (−0.372 − 0.645i)29-s + (−1.69 − 0.975i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.07220 - 0.394556i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.07220 - 0.394556i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 - 1.01i)T \) |
good | 5 | \( 1 + (-1.81 + 3.13i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (1.59 - 0.920i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-10.0 + 5.80i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-6.43 + 11.1i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 12.6T + 289T^{2} \) |
| 19 | \( 1 + 25.6iT - 361T^{2} \) |
| 23 | \( 1 + (-25.9 - 14.9i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (10.8 + 18.7i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (52.4 + 30.2i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 25.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (33.3 - 57.7i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-14.2 + 8.22i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-66.1 + 38.1i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 14.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-50.3 - 29.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (9.43 + 16.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-20.6 - 11.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 46.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 49.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + (52.4 - 30.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (86.2 - 49.8i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 154.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (21.1 + 36.5i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−12.78766031408470831583159386521, −11.60603894744029970576062542846, −10.89715394127719572289786198609, −9.597659792066135445444131355083, −8.894457828536137550426953702590, −7.20536064847101307819329990240, −5.92645790724763940563225747599, −5.13712866253830815454553799786, −3.58421512425179137672858490194, −0.968507342595558254918063923953,
1.62669349992560679914981584038, 3.81957729028955871820072175357, 5.41430279139394514522287634823, 6.58516944110582325904884349072, 7.20369602683073081771368411946, 8.934490441772733711123427213319, 10.18454736242680272387209434603, 10.90507240589649111908761629950, 12.05310510002668846160255500400, 12.70719122110043174674438803346