| L(s) = 1 | − 2·3-s + 4·7-s + 9-s + 12·13-s + 16·19-s − 8·21-s + 25-s + 4·27-s − 8·31-s − 4·37-s − 24·39-s − 4·43-s − 2·49-s − 32·57-s − 4·61-s + 4·63-s − 12·67-s + 20·73-s − 2·75-s + 16·79-s − 11·81-s + 48·91-s + 16·93-s + 20·97-s − 4·103-s + 28·109-s + 8·111-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s + 3.32·13-s + 3.67·19-s − 1.74·21-s + 1/5·25-s + 0.769·27-s − 1.43·31-s − 0.657·37-s − 3.84·39-s − 0.609·43-s − 2/7·49-s − 4.23·57-s − 0.512·61-s + 0.503·63-s − 1.46·67-s + 2.34·73-s − 0.230·75-s + 1.80·79-s − 1.22·81-s + 5.03·91-s + 1.65·93-s + 2.03·97-s − 0.394·103-s + 2.68·109-s + 0.759·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.369748295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.369748295\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−7.948156981393927508988974796934, −7.78598962640623119361177800709, −7.43941065459194807583635863256, −6.59250531831024987949367010419, −6.40764877261818687459848654265, −5.81408587465620821959066657575, −5.42261684437556226200264710088, −5.12441036610944036341041015255, −4.80343440735185988196356158458, −3.89787494876113843504859809365, −3.40336439874468655658501596905, −3.23319368204924250795089625252, −1.86143218348344746155921182825, −1.23403073899346067943514724338, −1.00571348946543017189879603649,
1.00571348946543017189879603649, 1.23403073899346067943514724338, 1.86143218348344746155921182825, 3.23319368204924250795089625252, 3.40336439874468655658501596905, 3.89787494876113843504859809365, 4.80343440735185988196356158458, 5.12441036610944036341041015255, 5.42261684437556226200264710088, 5.81408587465620821959066657575, 6.40764877261818687459848654265, 6.59250531831024987949367010419, 7.43941065459194807583635863256, 7.78598962640623119361177800709, 7.948156981393927508988974796934
