| L(s) = 1 | − 8.86·2-s + 1.50·3-s + 46.5·4-s − 37.8·5-s − 13.3·6-s − 124.·7-s − 129.·8-s − 240.·9-s + 335.·10-s + 590.·11-s + 69.9·12-s + 434.·13-s + 1.10e3·14-s − 56.8·15-s − 343.·16-s + 1.92e3·17-s + 2.13e3·18-s + 654.·19-s − 1.76e3·20-s − 187.·21-s − 5.23e3·22-s + 2.80e3·23-s − 194.·24-s − 1.69e3·25-s − 3.85e3·26-s − 726.·27-s − 5.81e3·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.0963·3-s + 1.45·4-s − 0.676·5-s − 0.150·6-s − 0.962·7-s − 0.714·8-s − 0.990·9-s + 1.06·10-s + 1.47·11-s + 0.140·12-s + 0.713·13-s + 1.50·14-s − 0.0651·15-s − 0.335·16-s + 1.61·17-s + 1.55·18-s + 0.415·19-s − 0.985·20-s − 0.0926·21-s − 2.30·22-s + 1.10·23-s − 0.0688·24-s − 0.542·25-s − 1.11·26-s − 0.191·27-s − 1.40·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6059024472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6059024472\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 + 8.86T + 32T^{2} \) |
| 3 | \( 1 - 1.50T + 243T^{2} \) |
| 5 | \( 1 + 37.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 124.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 590.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 434.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 654.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.97e3T + 1.15e8T^{2} \) |
| 47 | \( 1 - 2.20e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.84e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.88e4T + 8.58e9T^{2} \) |
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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
−15.34188352102271567701016646120, −13.93458291464995425414501630511, −12.08317128609614820257836666535, −11.16523757763141079317222780046, −9.711032950402186267292487581400, −8.856126538390184116956701603357, −7.66817218950546579436774081648, −6.27922955985001549821956599686, −3.39043826316152718059482802822, −0.872117925969098084697913757559,
0.872117925969098084697913757559, 3.39043826316152718059482802822, 6.27922955985001549821956599686, 7.66817218950546579436774081648, 8.856126538390184116956701603357, 9.711032950402186267292487581400, 11.16523757763141079317222780046, 12.08317128609614820257836666535, 13.93458291464995425414501630511, 15.34188352102271567701016646120
