| L(s) = 1 | + (−0.132 + 0.111i)2-s + (−0.342 + 1.94i)4-s + (−3.51 + 1.27i)5-s + (0.526 + 2.98i)7-s + (−0.343 − 0.594i)8-s + (0.323 − 0.559i)10-s + (−2.34 − 0.852i)11-s + (−0.586 − 0.491i)13-s + (−0.401 − 0.337i)14-s + (−3.59 − 1.30i)16-s + (2.31 − 4.00i)17-s + (0.305 + 0.529i)19-s + (−1.27 − 7.24i)20-s + (0.405 − 0.147i)22-s + (−1.13 + 6.42i)23-s + ⋯ |
| L(s) = 1 | + (−0.0936 + 0.0786i)2-s + (−0.171 + 0.970i)4-s + (−1.57 + 0.571i)5-s + (0.198 + 1.12i)7-s + (−0.121 − 0.210i)8-s + (0.102 − 0.177i)10-s + (−0.705 − 0.256i)11-s + (−0.162 − 0.136i)13-s + (−0.107 − 0.0900i)14-s + (−0.897 − 0.326i)16-s + (0.560 − 0.970i)17-s + (0.0701 + 0.121i)19-s + (−0.285 − 1.62i)20-s + (0.0863 − 0.0314i)22-s + (−0.236 + 1.33i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0883089 - 0.175837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0883089 - 0.175837i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.132 - 0.111i)T + (0.347 - 1.96i)T^{2} \) |
| 5 | \( 1 + (3.51 - 1.27i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.526 - 2.98i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.34 + 0.852i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.586 + 0.491i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.31 + 4.00i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.305 - 0.529i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.13 - 6.42i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.01 + 4.21i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.13 + 6.45i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.47 - 4.29i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.02 + 3.38i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (5.23 + 1.90i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.192 + 1.09i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 8.84T + 53T^{2} \) |
| 59 | \( 1 + (11.1 - 4.05i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.42 - 8.06i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.928 + 0.779i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.45 - 4.26i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.14 + 3.72i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.03 - 7.58i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.90 - 5.79i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.76 - 6.52i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.891 - 0.324i)T + (74.3 + 62.3i)T^{2} \) |
| 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
−11.30896471501111293974076068452, −10.05856750883833532005188128732, −9.005592655927878074849085221615, −8.051679995320003942590351724944, −7.80359524052834267776184731250, −6.88259538218554557449010182081, −5.51995350248192155220124347379, −4.44121612309901003969946054672, −3.33770436523774534867032863471, −2.70015059471794645954768134789,
0.11188077400779826054742399862, 1.37531704628227921978595179076, 3.35336836103518415788280650006, 4.54945345606469197930454788503, 4.87415692427833135991088280700, 6.39255634852619914736673103349, 7.34443334542174272793751288174, 8.158458498099617379683314905250, 8.808920862843851964202108244484, 10.18122902954633987110818388379
