| L(s) = 1 | + (−0.417 − 2.36i)2-s + (−0.210 + 1.71i)3-s + (−3.54 + 1.28i)4-s + (0.0713 + 0.0598i)5-s + (4.15 − 0.220i)6-s + (0.544 + 0.198i)7-s + (2.12 + 3.67i)8-s + (−2.91 − 0.722i)9-s + (0.111 − 0.193i)10-s + (−2.36 + 1.98i)11-s + (−1.47 − 6.35i)12-s + (0.729 − 4.13i)13-s + (0.241 − 1.37i)14-s + (−0.117 + 0.110i)15-s + (2.03 − 1.71i)16-s + (0.995 − 1.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.294 − 1.67i)2-s + (−0.121 + 0.992i)3-s + (−1.77 + 0.644i)4-s + (0.0318 + 0.0267i)5-s + (1.69 − 0.0898i)6-s + (0.205 + 0.0749i)7-s + (0.750 + 1.29i)8-s + (−0.970 − 0.240i)9-s + (0.0353 − 0.0612i)10-s + (−0.714 + 0.599i)11-s + (−0.424 − 1.83i)12-s + (0.202 − 1.14i)13-s + (0.0646 − 0.366i)14-s + (−0.0304 + 0.0284i)15-s + (0.509 − 0.427i)16-s + (0.241 − 0.418i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.456034 - 0.305088i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456034 - 0.305088i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.210 - 1.71i)T \) |
| good | 2 | \( 1 + (0.417 + 2.36i)T + (-1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-0.0713 - 0.0598i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.544 - 0.198i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (2.36 - 1.98i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.729 + 4.13i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.995 + 1.72i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.18 + 1.52i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 6.30i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.55 - 0.566i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (2.01 - 3.49i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.190 - 1.07i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.28 - 4.43i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (3.37 + 1.22i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 5.40T + 53T^{2} \) |
| 59 | \( 1 + (7.87 + 6.61i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-12.4 - 4.51i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.53 - 8.70i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.572 - 0.991i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.0977 + 0.169i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.25 + 7.09i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.58 + 14.6i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-0.776 - 1.34i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.05 + 3.40i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−17.64516651764500627543085725488, −16.07046329049329325618383054303, −14.64897852016930633147744427346, −13.01500691952281668963182732381, −11.81656664049646968336630172615, −10.57421682883284338796477237228, −9.910723521016106146839225707299, −8.428025627091345929483898172232, −4.99438736298409655793202273734, −3.14180862711550723881417911933,
5.40795628570150497123888584382, 6.76068675182425559962395746934, 7.86813441557953582351069766669, 9.072696096482557683720736346120, 11.37031216284053679685330909468, 13.25779482656238762043850678158, 14.05404275967019919056874332818, 15.36041552033424918467201891674, 16.61819122944113179353998044525, 17.42399262472809085136300549003
