| L(s) = 1 | − 2·3-s − 2·5-s + 9-s + 11-s + 4·13-s + 4·15-s − 17-s + 19-s − 3·23-s − 25-s + 4·27-s − 2·29-s + 4·31-s − 2·33-s − 4·37-s − 8·39-s + 2·41-s − 4·43-s − 2·45-s + 47-s + 2·51-s + 2·53-s − 2·55-s − 2·57-s − 4·59-s − 15·61-s − 8·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 1.03·15-s − 0.242·17-s + 0.229·19-s − 0.625·23-s − 1/5·25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s − 0.348·33-s − 0.657·37-s − 1.28·39-s + 0.312·41-s − 0.609·43-s − 0.298·45-s + 0.145·47-s + 0.280·51-s + 0.274·53-s − 0.269·55-s − 0.264·57-s − 0.520·59-s − 1.92·61-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−16.30389779934173, −15.84943343455857, −15.45644388440982, −14.83146918002910, −13.97494781577856, −13.61721682033615, −12.85323880083813, −12.05660840448536, −11.93504025024301, −11.33386620692942, −10.77975689076791, −10.40775381348832, −9.483934271960962, −8.867580740948050, −8.159915177579365, −7.704222283765665, −6.835190037247815, −6.311171030621713, −5.809906401970656, −5.065399481489468, −4.369148285795376, −3.757489733718989, −3.062288715423447, −1.835480675153686, −0.8844778525876288, 0,
0.8844778525876288, 1.835480675153686, 3.062288715423447, 3.757489733718989, 4.369148285795376, 5.065399481489468, 5.809906401970656, 6.311171030621713, 6.835190037247815, 7.704222283765665, 8.159915177579365, 8.867580740948050, 9.483934271960962, 10.40775381348832, 10.77975689076791, 11.33386620692942, 11.93504025024301, 12.05660840448536, 12.85323880083813, 13.61721682033615, 13.97494781577856, 14.83146918002910, 15.45644388440982, 15.84943343455857, 16.30389779934173
