| L(s) = 1 | − 2·3-s − 2·5-s − 9-s + 2·11-s + 4·15-s − 6·17-s − 10·19-s − 2·23-s + 25-s + 6·27-s + 14·31-s − 4·33-s − 6·37-s − 8·41-s − 8·43-s + 2·45-s + 18·47-s + 12·51-s + 2·53-s − 4·55-s + 20·57-s − 2·59-s − 6·61-s + 14·67-s + 4·69-s + 16·71-s − 18·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s − 1/3·9-s + 0.603·11-s + 1.03·15-s − 1.45·17-s − 2.29·19-s − 0.417·23-s + 1/5·25-s + 1.15·27-s + 2.51·31-s − 0.696·33-s − 0.986·37-s − 1.24·41-s − 1.21·43-s + 0.298·45-s + 2.62·47-s + 1.68·51-s + 0.274·53-s − 0.539·55-s + 2.64·57-s − 0.260·59-s − 0.768·61-s + 1.71·67-s + 0.481·69-s + 1.89·71-s − 2.10·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 21 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 61 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 29 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 14 T + 109 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 173 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 99 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 195 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 109 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 202 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.525869511495533364063787678443, −8.456755405680584718158436550752, −7.65210829654827355902817645938, −7.37845521750453703711904354387, −6.76854708126124718990297845499, −6.55845099979017514652587119119, −6.19294842713957101226758428952, −6.14781680915652749414094951160, −5.29532961046423438129161658160, −5.15741418476605100511842224937, −4.44019108543844189953525274728, −4.36590429000383657887290331704, −3.92856865693544620443884369723, −3.45835995508832654635027905907, −2.65669433272038829010243279193, −2.43269130072917020933100561140, −1.75193485107840393676975194688, −0.951957699095178357635460604610, 0, 0,
0.951957699095178357635460604610, 1.75193485107840393676975194688, 2.43269130072917020933100561140, 2.65669433272038829010243279193, 3.45835995508832654635027905907, 3.92856865693544620443884369723, 4.36590429000383657887290331704, 4.44019108543844189953525274728, 5.15741418476605100511842224937, 5.29532961046423438129161658160, 6.14781680915652749414094951160, 6.19294842713957101226758428952, 6.55845099979017514652587119119, 6.76854708126124718990297845499, 7.37845521750453703711904354387, 7.65210829654827355902817645938, 8.456755405680584718158436550752, 8.525869511495533364063787678443
