L(s) = 1 | − 2·3-s − 4-s + 9-s + 2·12-s − 3·16-s + 25-s + 4·27-s − 8·31-s − 36-s − 20·37-s + 6·48-s + 2·49-s + 7·64-s − 20·67-s − 2·75-s − 11·81-s + 16·93-s − 20·97-s − 100-s + 4·103-s − 4·108-s + 40·111-s − 11·121-s + 8·124-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1/2·4-s + 1/3·9-s + 0.577·12-s − 3/4·16-s + 1/5·25-s + 0.769·27-s − 1.43·31-s − 1/6·36-s − 3.28·37-s + 0.866·48-s + 2/7·49-s + 7/8·64-s − 2.44·67-s − 0.230·75-s − 1.22·81-s + 1.65·93-s − 2.03·97-s − 0.0999·100-s + 0.394·103-s − 0.384·108-s + 3.79·111-s − 121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−10.53816257725248073130842019726, −9.955951025311467647832971998211, −9.107317506658110308566345614657, −8.923644105979883940049846677080, −8.320437631237746381160261512250, −7.44997762456031466582066888485, −6.93370463159796136190684082007, −6.48980745991730713651759347225, −5.60065879084227609478701806218, −5.33377947704344434601290840834, −4.66473467177095454165541913899, −3.94727908945697743769298534363, −3.05940369593580445690460715793, −1.74717117060943513123928878837, 0,
1.74717117060943513123928878837, 3.05940369593580445690460715793, 3.94727908945697743769298534363, 4.66473467177095454165541913899, 5.33377947704344434601290840834, 5.60065879084227609478701806218, 6.48980745991730713651759347225, 6.93370463159796136190684082007, 7.44997762456031466582066888485, 8.320437631237746381160261512250, 8.923644105979883940049846677080, 9.107317506658110308566345614657, 9.955951025311467647832971998211, 10.53816257725248073130842019726