| L(s) = 1 | + 2·5-s − 2·9-s + 12·13-s + 4·17-s + 3·25-s + 4·29-s − 4·37-s − 20·41-s − 4·45-s − 10·49-s − 4·53-s − 4·61-s + 24·65-s + 20·73-s − 5·81-s + 8·85-s − 12·89-s + 20·97-s − 28·101-s + 28·109-s − 12·113-s − 24·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2/3·9-s + 3.32·13-s + 0.970·17-s + 3/5·25-s + 0.742·29-s − 0.657·37-s − 3.12·41-s − 0.596·45-s − 1.42·49-s − 0.549·53-s − 0.512·61-s + 2.97·65-s + 2.34·73-s − 5/9·81-s + 0.867·85-s − 1.27·89-s + 2.03·97-s − 2.78·101-s + 2.68·109-s − 1.12·113-s − 2.21·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.052262224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052262224\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.613228912214399908431984206107, −8.781404856404838385729722537924, −8.714543115468972113059546452375, −8.239480380770897255724570167213, −7.78598962640623119361177800709, −6.60040674980892955704654655808, −6.59250531831024987949367010419, −6.06254412327818625828699807195, −5.42261684437556226200264710088, −5.12441036610944036341041015255, −4.09416326053364009314265749502, −3.28314496469795666027926296042, −3.23319368204924250795089625252, −1.81188448910292297689023276847, −1.23403073899346067943514724338,
1.23403073899346067943514724338, 1.81188448910292297689023276847, 3.23319368204924250795089625252, 3.28314496469795666027926296042, 4.09416326053364009314265749502, 5.12441036610944036341041015255, 5.42261684437556226200264710088, 6.06254412327818625828699807195, 6.59250531831024987949367010419, 6.60040674980892955704654655808, 7.78598962640623119361177800709, 8.239480380770897255724570167213, 8.714543115468972113059546452375, 8.781404856404838385729722537924, 9.613228912214399908431984206107
