| L(s) = 1 | + 4-s − 6·5-s + 16-s − 6·20-s − 12·23-s + 18·25-s + 4·31-s − 49-s + 18·53-s − 12·59-s + 64-s − 12·67-s + 12·71-s − 6·80-s + 6·89-s − 12·92-s + 16·97-s + 18·100-s + 6·113-s + 72·115-s − 11·121-s + 4·124-s − 30·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 2.68·5-s + 1/4·16-s − 1.34·20-s − 2.50·23-s + 18/5·25-s + 0.718·31-s − 1/7·49-s + 2.47·53-s − 1.56·59-s + 1/8·64-s − 1.46·67-s + 1.42·71-s − 0.670·80-s + 0.635·89-s − 1.25·92-s + 1.62·97-s + 9/5·100-s + 0.564·113-s + 6.71·115-s − 121-s + 0.359·124-s − 2.68·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.61167421904303632670028187122, −7.42121673853141269825161473279, −6.81205900962282159354361821563, −6.33452929774039116014677262551, −6.00322867717322486820035996916, −5.36566179790692482996240876177, −4.74199771668790406151608318953, −4.33266919695477146669030135259, −3.85534020903213380080923841832, −3.73726778405345080419503026426, −3.09418209747352365991601981744, −2.48872174856063957758810279531, −1.79169260510919522904149861701, −0.75167882404979863467646960573, 0,
0.75167882404979863467646960573, 1.79169260510919522904149861701, 2.48872174856063957758810279531, 3.09418209747352365991601981744, 3.73726778405345080419503026426, 3.85534020903213380080923841832, 4.33266919695477146669030135259, 4.74199771668790406151608318953, 5.36566179790692482996240876177, 6.00322867717322486820035996916, 6.33452929774039116014677262551, 6.81205900962282159354361821563, 7.42121673853141269825161473279, 7.61167421904303632670028187122
