L(s) = 1 | + 3-s + 4·5-s − 7-s − 2·9-s + 2·13-s + 4·15-s + 3·19-s − 21-s + 4·23-s + 11·25-s − 5·27-s + 29-s − 9·31-s − 4·35-s − 10·37-s + 2·39-s − 8·41-s + 10·43-s − 8·45-s − 13·47-s + 49-s − 9·53-s + 3·57-s + 5·59-s + 8·61-s + 2·63-s + 8·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.377·7-s − 2/3·9-s + 0.554·13-s + 1.03·15-s + 0.688·19-s − 0.218·21-s + 0.834·23-s + 11/5·25-s − 0.962·27-s + 0.185·29-s − 1.61·31-s − 0.676·35-s − 1.64·37-s + 0.320·39-s − 1.24·41-s + 1.52·43-s − 1.19·45-s − 1.89·47-s + 1/7·49-s − 1.23·53-s + 0.397·57-s + 0.650·59-s + 1.02·61-s + 0.251·63-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.70223643538093, −13.34743068386163, −12.94738382608795, −12.50554867008534, −11.82273571029203, −11.08888373283651, −10.91462628978850, −10.17982819912020, −9.784116225012356, −9.310804142601651, −8.967442532605633, −8.546896677847593, −7.953133640918436, −7.209440262137308, −6.668139273116666, −6.345714961382676, −5.554427486482954, −5.378500404878143, −4.902472716036036, −3.793043927785175, −3.346291599698252, −2.874194328261207, −2.199548741407784, −1.716414728876964, −1.093675180856056, 0,
1.093675180856056, 1.716414728876964, 2.199548741407784, 2.874194328261207, 3.346291599698252, 3.793043927785175, 4.902472716036036, 5.378500404878143, 5.554427486482954, 6.345714961382676, 6.668139273116666, 7.209440262137308, 7.953133640918436, 8.546896677847593, 8.967442532605633, 9.310804142601651, 9.784116225012356, 10.17982819912020, 10.91462628978850, 11.08888373283651, 11.82273571029203, 12.50554867008534, 12.94738382608795, 13.34743068386163, 13.70223643538093