L(s) = 1 | − 2.05·2-s + 2.21·4-s + (0.0731 + 0.126i)5-s − 0.446·8-s + (−0.150 − 0.260i)10-s + (0.832 − 1.44i)11-s + (−0.0999 + 0.173i)13-s − 3.51·16-s + (3.13 + 5.43i)17-s + (−3.45 + 5.99i)19-s + (0.162 + 0.280i)20-s + (−1.70 + 2.95i)22-s + (−3.09 − 5.35i)23-s + (2.48 − 4.31i)25-s + (0.205 − 0.355i)26-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 1.10·4-s + (0.0327 + 0.0566i)5-s − 0.157·8-s + (−0.0474 − 0.0822i)10-s + (0.250 − 0.434i)11-s + (−0.0277 + 0.0480i)13-s − 0.879·16-s + (0.760 + 1.31i)17-s + (−0.793 + 1.37i)19-s + (0.0362 + 0.0627i)20-s + (−0.364 + 0.630i)22-s + (−0.644 − 1.11i)23-s + (0.497 − 0.862i)25-s + (0.0402 − 0.0697i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6302671443\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6302671443\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.05T + 2T^{2} \) |
| 5 | \( 1 + (-0.0731 - 0.126i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.832 + 1.44i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.0999 - 0.173i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.13 - 5.43i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.45 - 5.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.09 + 5.35i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.46 - 4.27i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 + (3.50 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.15 + 2.00i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.940 + 1.62i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 + (-2.67 - 4.62i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 - 0.678T + 61T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 + 1.27T + 71T^{2} \) |
| 73 | \( 1 + (-0.778 - 1.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + (-3.75 - 6.50i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.53 + 7.85i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.98 - 6.90i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.911877336921743199732601976272, −8.726811883280255251818078010629, −8.398688375363215644562147785457, −7.73651248022800801628278970051, −6.56581696544068233236385889824, −6.07602049340738436375151577801, −4.64521548277797799546686123443, −3.59182598490873726723020540075, −2.18000532291193180183887116813, −1.11362482477422804298866672951,
0.50319860117707937861137646698, 1.79451483593929747994991413696, 2.94034253910136035704973129829, 4.37845012543263320094190154398, 5.30069861150181125753121517471, 6.57415488789324037322812911998, 7.28429277360283634612344687008, 7.892626030991502558573919731250, 8.859209418579498965447647189777, 9.434152463482522389735548514408