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 | \( 1 + (-0.891 + 0.324i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.18122902954633987110818388379, −8.808920862843851964202108244484, −8.158458498099617379683314905250, −7.34443334542174272793751288174, −6.39255634852619914736673103349, −4.87415692427833135991088280700, −4.54945345606469197930454788503, −3.35336836103518415788280650006, −1.37531704628227921978595179076, −0.11188077400779826054742399862,
2.70015059471794645954768134789, 3.33770436523774534867032863471, 4.44121612309901003969946054672, 5.51995350248192155220124347379, 6.88259538218554557449010182081, 7.80359524052834267776184731250, 8.051679995320003942590351724944, 9.005592655927878074849085221615, 10.05856750883833532005188128732, 11.30896471501111293974076068452