L(s) = 1 | + (0.118 − 0.673i)2-s + (1.43 + 0.524i)4-s + (0.802 − 0.673i)5-s + (0.113 − 0.0412i)7-s + (1.20 − 2.09i)8-s + (−0.358 − 0.620i)10-s + (−4.16 − 3.49i)11-s + (−0.794 − 4.50i)13-s + (−0.0143 − 0.0812i)14-s + (1.08 + 0.907i)16-s + (−2.38 − 4.13i)17-s + (0.294 − 0.509i)19-s + (1.50 − 0.549i)20-s + (−2.84 + 2.38i)22-s + (7.32 + 2.66i)23-s + ⋯ |
L(s) = 1 | + (0.0839 − 0.476i)2-s + (0.719 + 0.262i)4-s + (0.359 − 0.301i)5-s + (0.0428 − 0.0155i)7-s + (0.427 − 0.739i)8-s + (−0.113 − 0.196i)10-s + (−1.25 − 1.05i)11-s + (−0.220 − 1.24i)13-s + (−0.00382 − 0.0217i)14-s + (0.270 + 0.226i)16-s + (−0.579 − 1.00i)17-s + (0.0675 − 0.116i)19-s + (0.337 − 0.122i)20-s + (−0.607 + 0.509i)22-s + (1.52 + 0.555i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42251 - 1.26593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42251 - 1.26593i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.118 + 0.673i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (-0.802 + 0.673i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.113 + 0.0412i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (4.16 + 3.49i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.794 + 4.50i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.38 + 4.13i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.294 + 0.509i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.32 - 2.66i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.880 + 4.99i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.23 - 2.99i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.09 - 1.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.31 - 7.44i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (1 + 0.839i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.27 - 0.826i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 + (0.0336 - 0.0282i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-9.59 + 3.49i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.322 + 1.83i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.25 - 5.63i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.11 - 10.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.121 + 0.691i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.17 + 6.67i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (3.42 - 5.93i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.91 + 5.80i)T + (16.8 + 95.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.30944256413542174623594352015, −9.570146870777693592366127694154, −8.337127422122223696163700531666, −7.70506024202875819792116362094, −6.69697666311433462081808272432, −5.61311025050166861941500784449, −4.81807969641304416818076977460, −3.11320807913761258177775993431, −2.71350252133016895899776913971, −0.966454563550970335119922011037,
1.89084738641495041672478950555, 2.67015658190047962158852866762, 4.45599997355821570822878007332, 5.24733945253350000671158137674, 6.43963999550525961659308901203, 6.88912371615983400908499040975, 7.81784695899641347579127120371, 8.767870317041011134806052257765, 9.952968839739956487829124766597, 10.52490400663501758622133312427