L(s) = 1 | + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)7-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)16-s + (0.5 − 0.866i)19-s + (0.173 − 0.984i)25-s − 28-s + (0.939 + 0.342i)31-s + (0.5 + 0.866i)37-s + (−0.766 − 0.642i)43-s + (−0.173 + 0.984i)52-s + (−1.87 + 0.684i)61-s + (−0.500 − 0.866i)64-s + (0.347 + 1.96i)67-s + (−1 + 1.73i)73-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)4-s + (0.939 − 0.342i)7-s + (−0.173 − 0.984i)13-s + (0.766 + 0.642i)16-s + (0.5 − 0.866i)19-s + (0.173 − 0.984i)25-s − 28-s + (0.939 + 0.342i)31-s + (0.5 + 0.866i)37-s + (−0.766 − 0.642i)43-s + (−0.173 + 0.984i)52-s + (−1.87 + 0.684i)61-s + (−0.500 − 0.866i)64-s + (0.347 + 1.96i)67-s + (−1 + 1.73i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8493336273\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8493336273\) |
\(L(1)\) |
|
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 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 11 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 13 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
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
−10.34579652374747927590146967109, −9.790311770213936977107662274657, −8.617491104520097752111525305875, −8.155167486862324897178058857488, −7.11394626090521797256455779443, −5.85019789089647864158040672938, −4.94697130196106096515591567895, −4.31598678520314317107371364768, −2.89788122762755178437030296667, −1.11039595924581354552842731452,
1.69316801652212957212058389030, 3.30006905457313040992511890378, 4.43165924689846831773873829699, 5.09415604745961991801379207341, 6.18459059752901128062402174739, 7.53903975070523437509804967260, 8.101609764971049034820272105992, 9.073948767431028297492851602817, 9.592040930190111692214303436407, 10.71343475221318595804443231018