| L(s) = 1 | − 2·3-s − 2·7-s + 9-s + 4·11-s + 8·17-s − 19-s + 4·21-s − 6·23-s + 4·27-s − 2·29-s + 8·31-s − 8·33-s − 6·41-s + 10·43-s + 6·47-s − 3·49-s − 16·51-s + 2·57-s − 4·59-s − 6·61-s − 2·63-s + 2·67-s + 12·69-s − 16·71-s − 16·73-s − 8·77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.94·17-s − 0.229·19-s + 0.872·21-s − 1.25·23-s + 0.769·27-s − 0.371·29-s + 1.43·31-s − 1.39·33-s − 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s − 2.24·51-s + 0.264·57-s − 0.520·59-s − 0.768·61-s − 0.251·63-s + 0.244·67-s + 1.44·69-s − 1.89·71-s − 1.87·73-s − 0.911·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.64912071845643, −14.68171783893831, −14.40527704918887, −13.84511423248265, −13.21145982660242, −12.42770584028955, −12.08116875029245, −11.87063269839820, −11.25171499929960, −10.38133749780879, −10.21311043931794, −9.571626149101406, −8.995399418149587, −8.284933863484774, −7.613130219655569, −7.004788448034921, −6.316700884620959, −5.923446674148174, −5.621803430898532, −4.646968037196384, −4.128503216855888, −3.371751382707999, −2.772982784568094, −1.568543084917359, −0.9324550840169829, 0,
0.9324550840169829, 1.568543084917359, 2.772982784568094, 3.371751382707999, 4.128503216855888, 4.646968037196384, 5.621803430898532, 5.923446674148174, 6.316700884620959, 7.004788448034921, 7.613130219655569, 8.284933863484774, 8.995399418149587, 9.571626149101406, 10.21311043931794, 10.38133749780879, 11.25171499929960, 11.87063269839820, 12.08116875029245, 12.42770584028955, 13.21145982660242, 13.84511423248265, 14.40527704918887, 14.68171783893831, 15.64912071845643
