| L(s) = 1 | − 32·2-s + 512·4-s + 3.75e3·5-s − 1.68e4·7-s − 1.20e5·10-s + 3.46e5·11-s − 4.65e5·13-s + 5.38e5·14-s − 2.62e5·16-s − 3.76e6·17-s + 1.92e6·20-s − 1.10e7·22-s + 1.04e7·23-s + 4.29e6·25-s + 1.48e7·26-s − 8.60e6·28-s − 2.01e7·31-s + 8.38e6·32-s + 1.20e8·34-s − 6.30e7·35-s + 1.12e8·37-s + 3.06e8·41-s + 1.18e8·43-s + 1.77e8·44-s − 3.34e8·46-s − 3.44e8·47-s + 1.41e8·49-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s + 6/5·5-s − 1.00·7-s − 6/5·10-s + 2.15·11-s − 1.25·13-s + 1.00·14-s − 1/4·16-s − 2.64·17-s + 3/5·20-s − 2.15·22-s + 1.62·23-s + 0.439·25-s + 1.25·26-s − 0.500·28-s − 0.703·31-s + 1/4·32-s + 2.64·34-s − 1.20·35-s + 1.62·37-s + 2.64·41-s + 0.807·43-s + 1.07·44-s − 1.62·46-s − 1.50·47-s + 0.500·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.085765872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.085765872\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 + p^{5} T + p^{9} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 6 p^{4} T + p^{10} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2402 p T + 2884802 p^{2} T^{2} + 2402 p^{11} T^{3} + p^{20} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 173398 T + p^{10} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 465246 T + 108226920258 T^{2} + 465246 p^{10} T^{3} + p^{20} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3760066 T + 7069048162178 T^{2} + 3760066 p^{10} T^{3} + p^{20} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 11048698082002 T^{2} + p^{20} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10456526 T + 54669467994338 T^{2} - 10456526 p^{10} T^{3} + p^{20} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 226828718694802 T^{2} + p^{20} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10065998 T + p^{10} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 112775826 T + 6359193464991138 T^{2} - 112775826 p^{10} T^{3} + p^{20} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 153003598 T + p^{10} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 118744914 T + 7050177300433698 T^{2} - 118744914 p^{10} T^{3} + p^{20} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 344678706 T + 59401705184917218 T^{2} + 344678706 p^{10} T^{3} + p^{20} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 139826 p^{2} T + 9775655138 p^{4} T^{2} + 139826 p^{12} T^{3} + p^{20} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 155272002554242 p^{2} T^{2} + p^{20} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 906185802 T + p^{10} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1924147934 T + 1851172635958234178 T^{2} + 1924147934 p^{10} T^{3} + p^{20} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3120877598 T + p^{10} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1272678526 T + 809855315270766338 T^{2} + 1272678526 p^{10} T^{3} + p^{20} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 15063541390202404802 T^{2} + p^{20} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10367644206 T + 53744023191102685218 T^{2} - 10367644206 p^{10} T^{3} + p^{20} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1969041775268808802 T^{2} + p^{20} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1280722494 T + 820125053318790018 T^{2} + 1280722494 p^{10} T^{3} + p^{20} T^{4} \) |
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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
−12.46991995141410288214899190978, −11.57328054895611207037768344024, −11.06284334688957871107084630113, −10.82271905800188502615220996772, −9.652603886119210251961421149327, −9.611899585498310887427886863467, −9.117989076421433583621586649147, −9.032244636271675686966930942909, −7.982427481322821424221255448225, −7.08495558030953894121985758589, −6.64083451701452249707127318826, −6.45346371608806624593173024231, −5.66354790357435491297412640257, −4.61802680315713853560045091243, −4.20856199032896179708230982553, −3.15990735681177531408575680899, −2.22851583328661518596239351337, −2.06164336272855706077561458301, −0.946436380490067268693730010612, −0.53438257236206558610103594612,
0.53438257236206558610103594612, 0.946436380490067268693730010612, 2.06164336272855706077561458301, 2.22851583328661518596239351337, 3.15990735681177531408575680899, 4.20856199032896179708230982553, 4.61802680315713853560045091243, 5.66354790357435491297412640257, 6.45346371608806624593173024231, 6.64083451701452249707127318826, 7.08495558030953894121985758589, 7.982427481322821424221255448225, 9.032244636271675686966930942909, 9.117989076421433583621586649147, 9.611899585498310887427886863467, 9.652603886119210251961421149327, 10.82271905800188502615220996772, 11.06284334688957871107084630113, 11.57328054895611207037768344024, 12.46991995141410288214899190978
