L(s) = 1 | − 2·5-s + 7-s − 3·9-s + 4·11-s + 2·13-s + 8·19-s − 25-s − 6·29-s − 8·31-s − 2·35-s + 2·37-s − 2·41-s − 4·43-s + 6·45-s − 8·47-s + 49-s + 6·53-s − 8·55-s + 6·61-s − 3·63-s − 4·65-s − 4·67-s + 8·71-s − 10·73-s + 4·77-s − 16·79-s + 9·81-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s + 1.83·19-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s + 0.824·53-s − 1.07·55-s + 0.768·61-s − 0.377·63-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.455·77-s − 1.80·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.31379751591723, −15.68554226219743, −15.05062830386625, −14.50036299341500, −14.23235751134799, −13.47689566204492, −12.96291329472491, −11.96179475833687, −11.75299005264246, −11.33522941382742, −10.90308444403268, −9.872760156609045, −9.360882311357562, −8.743434691715064, −8.272215088986851, −7.512507646111890, −7.166878636801463, −6.264244273616056, −5.607429359467706, −5.100050983280546, −4.102003853977522, −3.594493200325856, −3.103602357259613, −1.906791757334196, −1.105293326474413, 0,
1.105293326474413, 1.906791757334196, 3.103602357259613, 3.594493200325856, 4.102003853977522, 5.100050983280546, 5.607429359467706, 6.264244273616056, 7.166878636801463, 7.512507646111890, 8.272215088986851, 8.743434691715064, 9.360882311357562, 9.872760156609045, 10.90308444403268, 11.33522941382742, 11.75299005264246, 11.96179475833687, 12.96291329472491, 13.47689566204492, 14.23235751134799, 14.50036299341500, 15.05062830386625, 15.68554226219743, 16.31379751591723