| L(s) = 1 | + 2·3-s + 3·9-s − 2·11-s − 4·17-s − 6·25-s + 4·27-s − 4·33-s − 4·41-s + 2·49-s − 8·51-s + 8·59-s + 8·67-s − 28·73-s − 12·75-s + 5·81-s − 24·83-s − 12·89-s + 4·97-s − 6·99-s + 24·107-s − 12·113-s + 3·121-s − 8·123-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 0.603·11-s − 0.970·17-s − 6/5·25-s + 0.769·27-s − 0.696·33-s − 0.624·41-s + 2/7·49-s − 1.12·51-s + 1.04·59-s + 0.977·67-s − 3.27·73-s − 1.38·75-s + 5/9·81-s − 2.63·83-s − 1.27·89-s + 0.406·97-s − 0.603·99-s + 2.32·107-s − 1.12·113-s + 3/11·121-s − 0.721·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−7.75720782023225106841673661541, −7.57766902213530444274389318028, −6.91342543518502244816613622753, −6.87928263143909248442940323738, −5.90723738552130668805725208182, −5.79508255106269640718109448990, −5.08911955765620457569841389288, −4.35461089374668409702872860477, −4.32526881755629167390593741274, −3.53074178558790394542885743604, −3.10863384707893856832893049605, −2.45250023112523203194083612695, −2.07552052405763697992139591576, −1.33148151575350928602632304721, 0,
1.33148151575350928602632304721, 2.07552052405763697992139591576, 2.45250023112523203194083612695, 3.10863384707893856832893049605, 3.53074178558790394542885743604, 4.32526881755629167390593741274, 4.35461089374668409702872860477, 5.08911955765620457569841389288, 5.79508255106269640718109448990, 5.90723738552130668805725208182, 6.87928263143909248442940323738, 6.91342543518502244816613622753, 7.57766902213530444274389318028, 7.75720782023225106841673661541
