| L(s) = 1 | + (−0.744 − 0.373i)2-s + (−0.779 − 1.04i)4-s + (−0.345 − 1.15i)5-s + (−0.520 − 1.20i)7-s + (0.478 + 2.71i)8-s + (−0.174 + 0.989i)10-s + (−2.11 + 0.501i)11-s + (−3.80 − 2.50i)13-s + (−0.0636 + 1.09i)14-s + (−0.0912 + 0.304i)16-s + (−3.54 + 1.28i)17-s + (−2.50 − 0.911i)19-s + (−0.940 + 1.26i)20-s + (1.76 + 0.417i)22-s + (2.38 − 5.53i)23-s + ⋯ |
| L(s) = 1 | + (−0.526 − 0.264i)2-s + (−0.389 − 0.523i)4-s + (−0.154 − 0.516i)5-s + (−0.196 − 0.456i)7-s + (0.169 + 0.958i)8-s + (−0.0551 + 0.312i)10-s + (−0.637 + 0.151i)11-s + (−1.05 − 0.694i)13-s + (−0.0170 + 0.292i)14-s + (−0.0228 + 0.0761i)16-s + (−0.859 + 0.312i)17-s + (−0.574 − 0.209i)19-s + (−0.210 + 0.282i)20-s + (0.375 + 0.0890i)22-s + (0.498 − 1.15i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0637513 - 0.442932i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0637513 - 0.442932i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.744 + 0.373i)T + (1.19 + 1.60i)T^{2} \) |
| 5 | \( 1 + (0.345 + 1.15i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (0.520 + 1.20i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (2.11 - 0.501i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (3.80 + 2.50i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (3.54 - 1.28i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (2.50 + 0.911i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.38 + 5.53i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (0.241 + 4.15i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (7.40 - 0.865i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (-7.47 - 6.27i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (5.17 - 2.59i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (-3.46 - 3.67i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (2.97 + 0.348i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (-6.22 + 10.7i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.6 - 2.51i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (-7.04 + 9.45i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.0482 + 0.828i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (1.25 - 7.14i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.41 + 7.99i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (10.2 + 5.16i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (-4.26 - 2.14i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (0.578 + 3.28i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (0.390 - 1.30i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−11.44992429451169469862038690370, −10.47787006034921880269681142784, −9.902387482915817751870068995039, −8.773941802901010965241633773520, −8.029794062195454347093421516871, −6.69532767907624784963920651056, −5.24130719237228871948850251867, −4.42912893301757626854105498623, −2.40067448623759230486814773373, −0.41187327038587524857970224489,
2.60733428576145708950639834762, 4.00872252813001026546986001502, 5.38009365821098495047813152320, 6.97249275918529627644896323674, 7.47306851122723907269325789449, 8.820009311269348991481707700619, 9.363064093983987716641789213417, 10.54303188743624759151768163167, 11.55118029273202604522308949712, 12.62228070742874730615383945884
