| L(s) = 1 | + 2·5-s + 2·9-s + 8·13-s − 4·17-s − 25-s − 4·29-s + 12·37-s + 4·41-s + 4·45-s − 10·49-s + 8·61-s + 16·65-s − 4·73-s − 5·81-s − 8·85-s − 20·89-s + 28·97-s − 12·101-s − 16·113-s + 16·117-s + 2·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2/3·9-s + 2.21·13-s − 0.970·17-s − 1/5·25-s − 0.742·29-s + 1.97·37-s + 0.624·41-s + 0.596·45-s − 1.42·49-s + 1.02·61-s + 1.98·65-s − 0.468·73-s − 5/9·81-s − 0.867·85-s − 2.11·89-s + 2.84·97-s − 1.19·101-s − 1.50·113-s + 1.47·117-s + 2/11·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.664·145-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
| \(L(1)\) |
\(\approx\) |
\(2.065999675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.065999675\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
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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
−9.444819961268178302506198439876, −9.230125036457945551373457526789, −8.567608333828616047360588539462, −8.180394710372539090911010414030, −7.62020431866175871735973538240, −6.90119625539616123238624766672, −6.45878914610867592916836409873, −5.94814620683282854590317713202, −5.69109839538989913427593200609, −4.78050504695972043747613606224, −4.15350208143001890924072131336, −3.71915750418757685982538136223, −2.79549863875566012440421864841, −1.93108565620425138366120567805, −1.20865968811182505315883083764,
1.20865968811182505315883083764, 1.93108565620425138366120567805, 2.79549863875566012440421864841, 3.71915750418757685982538136223, 4.15350208143001890924072131336, 4.78050504695972043747613606224, 5.69109839538989913427593200609, 5.94814620683282854590317713202, 6.45878914610867592916836409873, 6.90119625539616123238624766672, 7.62020431866175871735973538240, 8.180394710372539090911010414030, 8.567608333828616047360588539462, 9.230125036457945551373457526789, 9.444819961268178302506198439876
