| L(s) = 1 | + (−0.548 + 1.83i)2-s + (−1.38 − 0.911i)4-s + (−0.984 + 2.28i)5-s + (0.264 + 4.54i)7-s + (−0.501 + 0.420i)8-s + (−3.64 − 3.05i)10-s + (−0.189 + 0.254i)11-s + (2.33 − 2.47i)13-s + (−8.47 − 2.00i)14-s + (−1.80 − 4.19i)16-s + (−0.885 + 5.02i)17-s + (−0.216 − 1.22i)19-s + (3.44 − 2.26i)20-s + (−0.362 − 0.487i)22-s + (−0.110 + 1.90i)23-s + ⋯ |
| L(s) = 1 | + (−0.387 + 1.29i)2-s + (−0.692 − 0.455i)4-s + (−0.440 + 1.02i)5-s + (0.100 + 1.71i)7-s + (−0.177 + 0.148i)8-s + (−1.15 − 0.965i)10-s + (−0.0572 + 0.0768i)11-s + (0.647 − 0.686i)13-s + (−2.26 − 0.536i)14-s + (−0.452 − 1.04i)16-s + (−0.214 + 1.21i)17-s + (−0.0495 − 0.281i)19-s + (0.769 − 0.506i)20-s + (−0.0773 − 0.103i)22-s + (−0.0231 + 0.397i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.433646 - 0.817657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.433646 - 0.817657i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.548 - 1.83i)T + (-1.67 - 1.09i)T^{2} \) |
| 5 | \( 1 + (0.984 - 2.28i)T + (-3.43 - 3.63i)T^{2} \) |
| 7 | \( 1 + (-0.264 - 4.54i)T + (-6.95 + 0.812i)T^{2} \) |
| 11 | \( 1 + (0.189 - 0.254i)T + (-3.15 - 10.5i)T^{2} \) |
| 13 | \( 1 + (-2.33 + 2.47i)T + (-0.755 - 12.9i)T^{2} \) |
| 17 | \( 1 + (0.885 - 5.02i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.216 + 1.22i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (0.110 - 1.90i)T + (-22.8 - 2.67i)T^{2} \) |
| 29 | \( 1 + (-3.37 + 0.799i)T + (25.9 - 13.0i)T^{2} \) |
| 31 | \( 1 + (-6.06 + 3.04i)T + (18.5 - 24.8i)T^{2} \) |
| 37 | \( 1 + (8.02 - 2.91i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (1.82 + 6.08i)T + (-34.2 + 22.5i)T^{2} \) |
| 43 | \( 1 + (-6.04 - 0.707i)T + (41.8 + 9.91i)T^{2} \) |
| 47 | \( 1 + (3.64 + 1.83i)T + (28.0 + 37.6i)T^{2} \) |
| 53 | \( 1 + (-3.88 + 6.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.37 + 3.18i)T + (-16.9 + 56.5i)T^{2} \) |
| 61 | \( 1 + (2.05 - 1.35i)T + (24.1 - 56.0i)T^{2} \) |
| 67 | \( 1 + (3.35 + 0.794i)T + (59.8 + 30.0i)T^{2} \) |
| 71 | \( 1 + (-2.16 - 1.81i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (3.24 - 2.71i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.07 + 6.94i)T + (-66.0 - 43.4i)T^{2} \) |
| 83 | \( 1 + (3.46 - 11.5i)T + (-69.3 - 45.6i)T^{2} \) |
| 89 | \( 1 + (2.86 - 2.40i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-5.79 - 13.4i)T + (-66.5 + 70.5i)T^{2} \) |
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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
−10.91586061261930633379453978873, −9.922952446045222565368978146761, −8.714876416666168760743520605337, −8.425512388809202669513479662061, −7.50267383498708770350486507967, −6.48273171562309124567266201627, −5.98527797905769482613932895158, −5.08557570588127053544601658561, −3.40834538620756422853334204314, −2.37256345519608723072600992949,
0.57911039504993471611111173910, 1.43484262641793270559793587544, 3.09511566388528546347196256888, 4.17303094188893824003274354133, 4.70297402353857215088479324328, 6.43913159160665779882848142924, 7.36236792060799456427081536473, 8.448578728342817956871324430244, 9.095302655967905806420808385168, 10.08192819033204173595837657075
