| L(s) = 1 | + (0.626 − 0.174i)2-s + (−1.34 + 0.814i)4-s + (−0.644 − 0.0250i)5-s + (−0.504 + 0.0789i)7-s + (−1.59 + 1.69i)8-s + (−0.408 + 0.0967i)10-s + (0.486 − 3.56i)11-s + (0.0982 + 0.143i)13-s + (−0.302 + 0.137i)14-s + (0.763 − 1.44i)16-s + (−6.29 − 3.16i)17-s + (−0.433 + 7.43i)19-s + (0.890 − 0.491i)20-s + (−0.316 − 2.31i)22-s + (−2.32 − 6.02i)23-s + ⋯ |
| L(s) = 1 | + (0.443 − 0.123i)2-s + (−0.674 + 0.407i)4-s + (−0.288 − 0.0111i)5-s + (−0.190 + 0.0298i)7-s + (−0.564 + 0.598i)8-s + (−0.129 + 0.0305i)10-s + (0.146 − 1.07i)11-s + (0.0272 + 0.0397i)13-s + (−0.0808 + 0.0367i)14-s + (0.190 − 0.362i)16-s + (−1.52 − 0.766i)17-s + (−0.0993 + 1.70i)19-s + (0.199 − 0.109i)20-s + (−0.0675 − 0.494i)22-s + (−0.485 − 1.25i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0533320 - 0.249591i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0533320 - 0.249591i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.626 + 0.174i)T + (1.71 - 1.03i)T^{2} \) |
| 5 | \( 1 + (0.644 + 0.0250i)T + (4.98 + 0.387i)T^{2} \) |
| 7 | \( 1 + (0.504 - 0.0789i)T + (6.66 - 2.13i)T^{2} \) |
| 11 | \( 1 + (-0.486 + 3.56i)T + (-10.5 - 2.94i)T^{2} \) |
| 13 | \( 1 + (-0.0982 - 0.143i)T + (-4.68 + 12.1i)T^{2} \) |
| 17 | \( 1 + (6.29 + 3.16i)T + (10.1 + 13.6i)T^{2} \) |
| 19 | \( 1 + (0.433 - 7.43i)T + (-18.8 - 2.20i)T^{2} \) |
| 23 | \( 1 + (2.32 + 6.02i)T + (-17.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (3.25 - 2.32i)T + (9.38 - 27.4i)T^{2} \) |
| 31 | \( 1 + (5.44 + 6.23i)T + (-4.19 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-3.71 + 4.98i)T + (-10.6 - 35.4i)T^{2} \) |
| 41 | \( 1 + (0.706 + 2.74i)T + (-35.8 + 19.8i)T^{2} \) |
| 43 | \( 1 + (5.41 - 4.90i)T + (4.16 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-5.10 + 5.84i)T + (-6.36 - 46.5i)T^{2} \) |
| 53 | \( 1 + (-1.26 - 7.14i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.549 + 0.224i)T + (42.1 + 41.3i)T^{2} \) |
| 61 | \( 1 + (-3.13 - 1.89i)T + (28.4 + 53.9i)T^{2} \) |
| 67 | \( 1 + (-5.50 - 3.93i)T + (21.6 + 63.3i)T^{2} \) |
| 71 | \( 1 + (3.45 - 11.5i)T + (-59.3 - 39.0i)T^{2} \) |
| 73 | \( 1 + (-8.73 - 2.06i)T + (65.2 + 32.7i)T^{2} \) |
| 79 | \( 1 + (-0.340 - 0.333i)T + (1.53 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-0.138 + 0.536i)T + (-72.6 - 40.0i)T^{2} \) |
| 89 | \( 1 + (3.77 + 12.6i)T + (-74.3 + 48.9i)T^{2} \) |
| 97 | \( 1 + (6.14 - 0.238i)T + (96.7 - 7.51i)T^{2} \) |
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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
−9.960703888948800896845548313302, −9.027265809385035310753350918648, −8.399905604726048084191497580918, −7.55997156718094570259106965920, −6.26000772805638777403955891730, −5.50228781998052158196020503264, −4.22836090567318874995035612437, −3.68944521889598781386244621502, −2.38036373936435666923225054329, −0.11028815781348492967772595371,
1.91622406007387393635620066521, 3.57542013082488548026686159724, 4.43736691781359599814301035766, 5.19832821626837996987389488160, 6.37017207593784699252076613421, 7.07886493334577602791454488335, 8.248634453067806567861192326967, 9.254459846057200006285561599145, 9.685417644184767542135813537966, 10.79576356849900422183126607767
