| L(s) = 1 | + (0.507 − 0.254i)2-s + (−1.00 + 1.34i)4-s + (0.832 − 2.78i)5-s + (1.23 − 2.87i)7-s + (−0.362 + 2.05i)8-s + (−0.286 − 1.62i)10-s + (3.33 + 0.791i)11-s + (3.33 − 2.19i)13-s + (−0.103 − 1.77i)14-s + (−0.622 − 2.07i)16-s + (−0.878 − 0.319i)17-s + (−4.55 + 1.65i)19-s + (2.90 + 3.90i)20-s + (1.89 − 0.449i)22-s + (2.43 + 5.64i)23-s + ⋯ |
| L(s) = 1 | + (0.358 − 0.180i)2-s + (−0.500 + 0.672i)4-s + (0.372 − 1.24i)5-s + (0.468 − 1.08i)7-s + (−0.128 + 0.726i)8-s + (−0.0905 − 0.513i)10-s + (1.00 + 0.238i)11-s + (0.924 − 0.607i)13-s + (−0.0276 − 0.473i)14-s + (−0.155 − 0.519i)16-s + (−0.213 − 0.0775i)17-s + (−1.04 + 0.380i)19-s + (0.650 + 0.873i)20-s + (0.404 − 0.0957i)22-s + (0.508 + 1.17i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39726 - 0.525560i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39726 - 0.525560i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.507 + 0.254i)T + (1.19 - 1.60i)T^{2} \) |
| 5 | \( 1 + (-0.832 + 2.78i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (-1.23 + 2.87i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (-3.33 - 0.791i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (-3.33 + 2.19i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (0.878 + 0.319i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (4.55 - 1.65i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-2.43 - 5.64i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (-0.104 + 1.79i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (0.671 + 0.0784i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (8.73 - 7.32i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (5.94 + 2.98i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.62i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (3.14 - 0.367i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (-4.18 - 7.25i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.89 + 1.16i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (0.340 + 0.457i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.794 - 13.6i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (-2.31 - 13.1i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.17 - 6.64i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.666 + 0.334i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (-3.25 + 1.63i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (-2.27 + 12.9i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (3.58 + 11.9i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−12.15371004145191760046160333302, −11.25950847084003081758303881400, −10.04618027908505495445485378914, −8.836740262456177997879815877968, −8.358739732135370649624392214690, −7.08765079320001695160825560429, −5.52604983999380736477534647198, −4.46340290660642519487855917379, −3.68227069241778774485627453391, −1.37085876790033961353053296718,
2.02491174216705621214787036364, 3.70018898174577743970719729953, 5.05561315590548288859969001181, 6.36641113412314444024678740291, 6.60091405660315344528087868064, 8.641037934208111287073145831033, 9.155441611164351356055148937425, 10.50444061795112937219077444026, 11.09122720876594161546442548098, 12.21753163144831849135182563178
