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
\(L(1)\) |
\(\approx\) |
\(-1.03219i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-1.03219i\) |
\(L(\frac{3}{2})\) |
|
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
−10.95450235084996937674413289009, −9.650337126914099419336180987509, −8.967834974027086461177210431995, −8.379810723645269485083599518101, −7.16988959514259210232032775274, −6.31114851557478658734128954283, −6.15042213490454612861795348314, −4.43309039349044947179248004439, −3.12456900298000725644776433212, −2.22192789381039815608951698888,
0.57693279686071007175702932035, 1.79643752401356964227417481285, 3.11281594885382635757352578998, 4.34377026256273973272644082075, 5.53369000776621485392560280703, 6.37277920403888121652973632023, 7.31626692163085868640353603372, 8.611624232022514186331161618235, 9.239222456721195254313498311829, 10.05635863512140669046883812691