| L(s) = 1 | − 2·3-s − 4·4-s + 3·9-s + 8·12-s + 12·16-s − 4·19-s − 10·27-s − 12·36-s + 40·43-s − 24·48-s + 28·49-s + 8·57-s − 32·64-s + 28·67-s + 4·73-s + 16·76-s + 20·81-s + 40·97-s + 40·108-s − 14·121-s + 127-s − 80·129-s + 131-s + 137-s + 139-s + 36·144-s − 56·147-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2·4-s + 9-s + 2.30·12-s + 3·16-s − 0.917·19-s − 1.92·27-s − 2·36-s + 6.09·43-s − 3.46·48-s + 4·49-s + 1.05·57-s − 4·64-s + 3.42·67-s + 0.468·73-s + 1.83·76-s + 20/9·81-s + 4.06·97-s + 3.84·108-s − 1.27·121-s + 0.0887·127-s − 7.04·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3·144-s − 4.61·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.073571789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.073571789\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.68198681095650743273350124269, −7.60135815847901278460459604886, −7.23452383498496106765582038210, −6.95604292129274155874022633704, −6.81609947955012021188071711995, −6.39494562166137588659466204729, −6.02114194011809143698325143771, −5.83492678157565143534345987904, −5.68472129577490533157520714386, −5.60323674393187027476246252548, −5.42104755747051346739970064997, −4.92337970826282915247093531644, −4.60436787677055530276936602256, −4.54193481752522462504601853870, −4.11122225182346979569168121830, −4.07720853240452779954542107476, −3.81000522356168692752371828353, −3.54569085142648114638685325028, −3.13996614076996600979206298663, −2.37672987597234901674568978512, −2.36544737574726336041860802344, −2.04832283639156490265104378621, −1.05974494818868922471617885363, −0.805908420692261918825522403684, −0.58663908602624407808599996061,
0.58663908602624407808599996061, 0.805908420692261918825522403684, 1.05974494818868922471617885363, 2.04832283639156490265104378621, 2.36544737574726336041860802344, 2.37672987597234901674568978512, 3.13996614076996600979206298663, 3.54569085142648114638685325028, 3.81000522356168692752371828353, 4.07720853240452779954542107476, 4.11122225182346979569168121830, 4.54193481752522462504601853870, 4.60436787677055530276936602256, 4.92337970826282915247093531644, 5.42104755747051346739970064997, 5.60323674393187027476246252548, 5.68472129577490533157520714386, 5.83492678157565143534345987904, 6.02114194011809143698325143771, 6.39494562166137588659466204729, 6.81609947955012021188071711995, 6.95604292129274155874022633704, 7.23452383498496106765582038210, 7.60135815847901278460459604886, 7.68198681095650743273350124269
