| L(s) = 1 | + 2·5-s − 10·13-s + 16·17-s + 2·25-s − 6·29-s + 14·37-s + 14·49-s − 18·53-s + 22·61-s − 20·65-s − 9·81-s + 32·85-s − 16·97-s − 18·101-s + 26·109-s − 28·113-s + 10·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.77·13-s + 3.88·17-s + 2/5·25-s − 1.11·29-s + 2.30·37-s + 2·49-s − 2.47·53-s + 2.81·61-s − 2.48·65-s − 81-s + 3.47·85-s − 1.62·97-s − 1.79·101-s + 2.49·109-s − 2.63·113-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.044004972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.044004972\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−11.06236775639795567100244089205, −10.52001374860430329285730054128, −9.896753913495995417310259114373, −9.758859936295986518668239176825, −9.714459441067935872421169197455, −9.196400020440862872369552590347, −8.233111594885068046670183703719, −7.908003555302383178461448082083, −7.45643673119267695566580024035, −7.26818589013450089758301648273, −6.54334900405889425656547510256, −5.77660705467302345136684631443, −5.38909910079939966506619296906, −5.36316779682565399323150483515, −4.51558208219417136938845621403, −3.85485476713265904517512018193, −2.97598496441294941622438905756, −2.69885519437037653529796435207, −1.85044050054771805168019821255, −0.892281956705227384142959632513,
0.892281956705227384142959632513, 1.85044050054771805168019821255, 2.69885519437037653529796435207, 2.97598496441294941622438905756, 3.85485476713265904517512018193, 4.51558208219417136938845621403, 5.36316779682565399323150483515, 5.38909910079939966506619296906, 5.77660705467302345136684631443, 6.54334900405889425656547510256, 7.26818589013450089758301648273, 7.45643673119267695566580024035, 7.908003555302383178461448082083, 8.233111594885068046670183703719, 9.196400020440862872369552590347, 9.714459441067935872421169197455, 9.758859936295986518668239176825, 9.896753913495995417310259114373, 10.52001374860430329285730054128, 11.06236775639795567100244089205
