| L(s) = 1 | − 2-s − 3·3-s − 2·4-s − 3·5-s + 3·6-s + 3·7-s + 3·8-s + 2·9-s + 3·10-s + 6·12-s + 13-s − 3·14-s + 9·15-s + 16-s − 4·17-s − 2·18-s + 2·19-s + 6·20-s − 9·21-s + 4·23-s − 9·24-s − 2·25-s − 26-s + 6·27-s − 6·28-s − 18·29-s − 9·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 4-s − 1.34·5-s + 1.22·6-s + 1.13·7-s + 1.06·8-s + 2/3·9-s + 0.948·10-s + 1.73·12-s + 0.277·13-s − 0.801·14-s + 2.32·15-s + 1/4·16-s − 0.970·17-s − 0.471·18-s + 0.458·19-s + 1.34·20-s − 1.96·21-s + 0.834·23-s − 1.83·24-s − 2/5·25-s − 0.196·26-s + 1.15·27-s − 1.13·28-s − 3.34·29-s − 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32341969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32341969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | | \( 1 \) |
| 47 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 25 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 91 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 135 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 123 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 113 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 125 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 173 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 239 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 149 T^{2} + p^{2} T^{4} \) |
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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
−7.69889952025520372874330286261, −7.67756283257147285951129414602, −7.31769741387222476595669474727, −7.21961423699867321786264827194, −6.44824850245512109269483857828, −5.95326208547498890982616152123, −5.80769016271127479096417125839, −5.46784837018212606625218687679, −4.89986582765225426165279787445, −4.85988956032325708922026743063, −4.27759773965262635427123819436, −4.17464832663752781873318726689, −3.43462598153875759772487613602, −3.42435789328946906373866063059, −2.39068489842424337610336784062, −1.86140459426729765786698715559, −1.30369719390986037188790653672, −0.67370676791448655064115416783, 0, 0,
0.67370676791448655064115416783, 1.30369719390986037188790653672, 1.86140459426729765786698715559, 2.39068489842424337610336784062, 3.42435789328946906373866063059, 3.43462598153875759772487613602, 4.17464832663752781873318726689, 4.27759773965262635427123819436, 4.85988956032325708922026743063, 4.89986582765225426165279787445, 5.46784837018212606625218687679, 5.80769016271127479096417125839, 5.95326208547498890982616152123, 6.44824850245512109269483857828, 7.21961423699867321786264827194, 7.31769741387222476595669474727, 7.67756283257147285951129414602, 7.69889952025520372874330286261
