| L(s) = 1 | + (0.940 + 0.147i)2-s + (−1.04 − 0.333i)4-s + (−0.357 − 0.162i)5-s + (−0.150 − 1.10i)7-s + (−2.63 − 1.32i)8-s + (−0.311 − 0.205i)10-s + (−4.62 + 0.359i)11-s + (1.29 + 3.78i)13-s + (0.0205 − 1.05i)14-s + (−0.501 − 0.358i)16-s + (−0.303 + 0.703i)17-s + (−3.78 + 5.09i)19-s + (0.317 + 0.288i)20-s + (−4.40 − 0.342i)22-s + (−3.42 − 2.65i)23-s + ⋯ |
| L(s) = 1 | + (0.665 + 0.104i)2-s + (−0.520 − 0.166i)4-s + (−0.159 − 0.0725i)5-s + (−0.0568 − 0.416i)7-s + (−0.930 − 0.467i)8-s + (−0.0986 − 0.0648i)10-s + (−1.39 + 0.108i)11-s + (0.358 + 1.04i)13-s + (0.00548 − 0.282i)14-s + (−0.125 − 0.0895i)16-s + (−0.0735 + 0.170i)17-s + (−0.869 + 1.16i)19-s + (0.0710 + 0.0644i)20-s + (−0.938 − 0.0729i)22-s + (−0.713 − 0.553i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00268436 + 0.0399218i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00268436 + 0.0399218i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.940 - 0.147i)T + (1.90 + 0.610i)T^{2} \) |
| 5 | \( 1 + (0.357 + 0.162i)T + (3.28 + 3.76i)T^{2} \) |
| 7 | \( 1 + (0.150 + 1.10i)T + (-6.74 + 1.87i)T^{2} \) |
| 11 | \( 1 + (4.62 - 0.359i)T + (10.8 - 1.69i)T^{2} \) |
| 13 | \( 1 + (-1.29 - 3.78i)T + (-10.2 + 7.96i)T^{2} \) |
| 17 | \( 1 + (0.303 - 0.703i)T + (-11.6 - 12.3i)T^{2} \) |
| 19 | \( 1 + (3.78 - 5.09i)T + (-5.44 - 18.2i)T^{2} \) |
| 23 | \( 1 + (3.42 + 2.65i)T + (5.73 + 22.2i)T^{2} \) |
| 29 | \( 1 + (6.49 - 3.92i)T + (13.5 - 25.6i)T^{2} \) |
| 31 | \( 1 + (1.45 - 0.0565i)T + (30.9 - 2.40i)T^{2} \) |
| 37 | \( 1 + (-2.67 + 2.83i)T + (-2.15 - 36.9i)T^{2} \) |
| 41 | \( 1 + (-0.408 + 1.05i)T + (-30.3 - 27.5i)T^{2} \) |
| 43 | \( 1 + (2.72 + 10.5i)T + (-37.6 + 20.7i)T^{2} \) |
| 47 | \( 1 + (5.25 + 0.203i)T + (46.8 + 3.64i)T^{2} \) |
| 53 | \( 1 + (-11.9 + 4.34i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-3.80 - 7.96i)T + (-37.0 + 45.9i)T^{2} \) |
| 61 | \( 1 + (8.01 - 2.56i)T + (49.6 - 35.4i)T^{2} \) |
| 67 | \( 1 + (9.11 + 5.50i)T + (31.2 + 59.2i)T^{2} \) |
| 71 | \( 1 + (0.0981 - 1.68i)T + (-70.5 - 8.24i)T^{2} \) |
| 73 | \( 1 + (5.22 - 3.43i)T + (28.9 - 67.0i)T^{2} \) |
| 79 | \( 1 + (-5.04 + 6.25i)T + (-16.7 - 77.2i)T^{2} \) |
| 83 | \( 1 + (-1.55 - 4.02i)T + (-61.4 + 55.7i)T^{2} \) |
| 89 | \( 1 + (-0.222 - 3.82i)T + (-88.3 + 10.3i)T^{2} \) |
| 97 | \( 1 + (-2.89 + 1.31i)T + (63.7 - 73.0i)T^{2} \) |
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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
−10.58093358003202776125472623310, −10.15592612933380738433875247667, −9.006583057178489926061471014896, −8.278520869358630469713700133467, −7.26420671338965592167947971009, −6.14752545153076376665107640143, −5.41688387277712506457562369006, −4.28253739258508802630667813956, −3.74083290302858538274714617661, −2.11225725955840888751159301513,
0.01569082743070994623629200247, 2.48039362598974734134996573456, 3.36340300526494117669973716566, 4.50523035788943289178789699995, 5.43328577703572746552451042731, 6.02991446046760687893327554523, 7.57620447004348711772010899448, 8.185096038660589632635540420129, 9.106513302764984830927521583950, 9.980219958795649179647750116729
