| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s − 5·7-s + 4·8-s + 9-s − 3·12-s − 10·14-s + 5·16-s + 2·18-s + 19-s + 5·21-s − 4·24-s − 25-s − 27-s − 15·28-s + 3·29-s + 6·32-s + 3·36-s + 2·38-s − 9·41-s + 10·42-s + 43-s − 5·48-s + 7·49-s − 2·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s − 1.88·7-s + 1.41·8-s + 1/3·9-s − 0.866·12-s − 2.67·14-s + 5/4·16-s + 0.471·18-s + 0.229·19-s + 1.09·21-s − 0.816·24-s − 1/5·25-s − 0.192·27-s − 2.83·28-s + 0.557·29-s + 1.06·32-s + 1/2·36-s + 0.324·38-s − 1.40·41-s + 1.54·42-s + 0.152·43-s − 0.721·48-s + 49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
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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
−7.895066880100714880010765073895, −7.37537050598211728364108602982, −6.79640354020912834681147009410, −6.65100909763992877932040013392, −6.16797338103572917911190248799, −5.93044412064917709044791732692, −5.16664875685110970250235628594, −5.00645295207967492451438634426, −4.29167327291336855211580789425, −3.73710485726989622889330712148, −3.34819909146828873634094986391, −2.89950449016663250242264949545, −2.23325747212457775082930091172, −1.28939472445932661570251507677, 0,
1.28939472445932661570251507677, 2.23325747212457775082930091172, 2.89950449016663250242264949545, 3.34819909146828873634094986391, 3.73710485726989622889330712148, 4.29167327291336855211580789425, 5.00645295207967492451438634426, 5.16664875685110970250235628594, 5.93044412064917709044791732692, 6.16797338103572917911190248799, 6.65100909763992877932040013392, 6.79640354020912834681147009410, 7.37537050598211728364108602982, 7.895066880100714880010765073895
