| L(s) = 1 | − 4·3-s + 6·9-s + 4·11-s − 4·17-s − 4·19-s − 6·25-s + 4·27-s − 16·33-s − 12·41-s − 12·43-s + 2·49-s + 16·51-s + 16·57-s − 28·59-s − 20·67-s + 28·73-s + 24·75-s − 37·81-s + 12·83-s − 4·89-s − 4·97-s + 24·99-s + 4·107-s + 4·113-s − 10·121-s + 48·123-s + 127-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s + 1.20·11-s − 0.970·17-s − 0.917·19-s − 6/5·25-s + 0.769·27-s − 2.78·33-s − 1.87·41-s − 1.82·43-s + 2/7·49-s + 2.24·51-s + 2.11·57-s − 3.64·59-s − 2.44·67-s + 3.27·73-s + 2.77·75-s − 4.11·81-s + 1.31·83-s − 0.423·89-s − 0.406·97-s + 2.41·99-s + 0.386·107-s + 0.376·113-s − 0.909·121-s + 4.32·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−10.90944990278511493765148592647, −10.57265847830629711938727340297, −9.886348706897054809183790479328, −9.215939702261382322141825805489, −8.635939205168057076823393046651, −7.984207807149970007419508597610, −6.97464122003923989664965494039, −6.39871498618546548325782647590, −6.33050260025483473718379487161, −5.59100833138318705450052051530, −4.84373431553687523132526670248, −4.43324404972048918127214240916, −3.36730269661969347724618284285, −1.73703685485230483844290438322, 0,
1.73703685485230483844290438322, 3.36730269661969347724618284285, 4.43324404972048918127214240916, 4.84373431553687523132526670248, 5.59100833138318705450052051530, 6.33050260025483473718379487161, 6.39871498618546548325782647590, 6.97464122003923989664965494039, 7.984207807149970007419508597610, 8.635939205168057076823393046651, 9.215939702261382322141825805489, 9.886348706897054809183790479328, 10.57265847830629711938727340297, 10.90944990278511493765148592647
