| L(s) = 1 | + 2-s + 4-s + 4·5-s + 8-s + 4·10-s − 8·13-s + 16-s + 4·20-s + 2·25-s − 8·26-s − 4·29-s + 32-s − 16·37-s + 4·40-s − 4·41-s − 14·49-s + 2·50-s − 8·52-s + 4·53-s − 4·58-s − 20·61-s + 64-s − 32·65-s + 12·73-s − 16·74-s + 4·80-s − 4·82-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s + 1.26·10-s − 2.21·13-s + 1/4·16-s + 0.894·20-s + 2/5·25-s − 1.56·26-s − 0.742·29-s + 0.176·32-s − 2.63·37-s + 0.632·40-s − 0.624·41-s − 2·49-s + 0.282·50-s − 1.10·52-s + 0.549·53-s − 0.525·58-s − 2.56·61-s + 1/8·64-s − 3.96·65-s + 1.40·73-s − 1.85·74-s + 0.447·80-s − 0.441·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | | \( 1 \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.72097144375492209353190700554, −7.43472461846693450494480621223, −6.92076782170002683072135481731, −6.59001220829617312734896517440, −6.04537670491531984013901490450, −5.53295861295123136716745752833, −5.39006577777807946122287060730, −4.65486938055172924173421055502, −4.64180517544954192308318972299, −3.53711099784593745581971184422, −3.17827142125620977279897528792, −2.45080129749770972997412968234, −1.85086593928300837816640591535, −1.75738623548621053409146805722, 0,
1.75738623548621053409146805722, 1.85086593928300837816640591535, 2.45080129749770972997412968234, 3.17827142125620977279897528792, 3.53711099784593745581971184422, 4.64180517544954192308318972299, 4.65486938055172924173421055502, 5.39006577777807946122287060730, 5.53295861295123136716745752833, 6.04537670491531984013901490450, 6.59001220829617312734896517440, 6.92076782170002683072135481731, 7.43472461846693450494480621223, 7.72097144375492209353190700554
