L(s) = 1 | + (0.527 + 0.913i)2-s + (0.444 − 0.769i)4-s + (−0.872 + 1.51i)5-s + (−1.22 − 2.12i)7-s + 3.04·8-s − 1.84·10-s + (−0.627 − 1.08i)11-s + (2.27 − 3.93i)13-s + (1.29 − 2.24i)14-s + (0.716 + 1.24i)16-s + 6.64·17-s + 0.249·19-s + (0.775 + 1.34i)20-s + (0.661 − 1.14i)22-s + (0.421 − 0.729i)23-s + ⋯ |
L(s) = 1 | + (0.372 + 0.645i)2-s + (0.222 − 0.384i)4-s + (−0.390 + 0.676i)5-s + (−0.464 − 0.804i)7-s + 1.07·8-s − 0.582·10-s + (−0.189 − 0.327i)11-s + (0.630 − 1.09i)13-s + (0.346 − 0.600i)14-s + (0.179 + 0.310i)16-s + 1.61·17-s + 0.0571·19-s + (0.173 + 0.300i)20-s + (0.140 − 0.244i)22-s + (0.0877 − 0.152i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.94284\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94284\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.527 - 0.913i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.872 - 1.51i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.22 + 2.12i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.627 + 1.08i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.27 + 3.93i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 - 0.249T + 19T^{2} \) |
| 23 | \( 1 + (-0.421 + 0.729i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.256 + 0.443i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.410 + 0.710i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.60T + 37T^{2} \) |
| 41 | \( 1 + (4.07 - 7.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.16 + 3.74i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.65 + 4.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + (1.50 - 2.60i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.44 + 2.49i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.04 - 8.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.0894T + 71T^{2} \) |
| 73 | \( 1 + 5.32T + 73T^{2} \) |
| 79 | \( 1 + (2.38 + 4.13i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.02 - 6.96i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 6.70T + 89T^{2} \) |
| 97 | \( 1 + (2.74 + 4.75i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.37714649869056788087183882824, −9.891983827606449945477682107941, −8.321224476799075994622354158649, −7.54800459229597046412623287217, −6.91244212461265926611469919440, −5.96455191351103100274091807709, −5.24040589379878263546124770860, −3.85487489285598910773267762256, −3.02762033369316650543083591248, −1.03631284188297956351457445142,
1.50551843311531596567366434274, 2.80852283227758104415862878486, 3.78698566853830306911264779796, 4.71772576718729526204611515392, 5.79416917176819939554188365962, 6.94335526412476654708273487210, 7.903232929122744651234653210263, 8.694495967756963671315827311996, 9.571489413350616433099430077542, 10.52071594402582512014035386141