L(s) = 1 | + (−1.19 + 0.687i)5-s + (−1.80 + 3.12i)7-s + (−1.83 − 1.05i)11-s + (0.887 − 0.512i)13-s − 0.808·17-s − 7.43i·19-s + (−1.65 − 2.86i)23-s + (−1.55 + 2.69i)25-s + (−7.71 − 4.45i)29-s + (−3.26 − 5.65i)31-s − 4.96i·35-s − 4.01i·37-s + (3.45 + 5.99i)41-s + (−0.245 − 0.142i)43-s + (−3.61 + 6.25i)47-s + ⋯ |
L(s) = 1 | + (−0.532 + 0.307i)5-s + (−0.682 + 1.18i)7-s + (−0.552 − 0.319i)11-s + (0.246 − 0.142i)13-s − 0.196·17-s − 1.70i·19-s + (−0.345 − 0.597i)23-s + (−0.310 + 0.538i)25-s + (−1.43 − 0.826i)29-s + (−0.586 − 1.01i)31-s − 0.839i·35-s − 0.660i·37-s + (0.540 + 0.935i)41-s + (−0.0375 − 0.0216i)43-s + (−0.527 + 0.912i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0315886 - 0.110145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0315886 - 0.110145i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.19 - 0.687i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.80 - 3.12i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.83 + 1.05i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.887 + 0.512i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 0.808T + 17T^{2} \) |
| 19 | \( 1 + 7.43iT - 19T^{2} \) |
| 23 | \( 1 + (1.65 + 2.86i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (7.71 + 4.45i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.26 + 5.65i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.01iT - 37T^{2} \) |
| 41 | \( 1 + (-3.45 - 5.99i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.245 + 0.142i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.61 - 6.25i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3.86iT - 53T^{2} \) |
| 59 | \( 1 + (7.06 - 4.08i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.31 + 3.64i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.43 - 1.40i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.69T + 71T^{2} \) |
| 73 | \( 1 - 0.409T + 73T^{2} \) |
| 79 | \( 1 + (-0.0456 + 0.0790i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.40 - 1.39i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 8.91T + 89T^{2} \) |
| 97 | \( 1 + (2.76 - 4.78i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.536529048830253126721641300849, −9.158203896336357099833970432870, −8.099440247052041147186350837875, −7.32368156713824574280074548541, −6.22548742528837927547303436199, −5.57916638307263351719562290921, −4.35947018676152275785799536890, −3.15844864847558559863126434535, −2.34615627980462681483183809548, −0.05306418794642654693044879782,
1.65960262315978523026004776507, 3.46645948875768917687518319888, 3.98481308947411831262663629972, 5.18211496993334971919440568121, 6.24835539250058333534048306760, 7.27526911715582307127577751888, 7.80922246285338578546170027108, 8.794069098149352501171498001384, 9.859804025210485223108286255050, 10.39121062113890255024699836158