| L(s) = 1 | + 366·3-s − 3.75e3·5-s + 1.68e4·7-s + 6.69e4·9-s + 3.46e5·11-s − 4.65e5·13-s − 1.37e6·15-s + 3.76e6·17-s + 6.15e6·21-s + 1.04e7·23-s + 4.29e6·25-s + 2.16e7·27-s + 2.01e7·31-s + 1.26e8·33-s − 6.30e7·35-s + 1.12e8·37-s − 1.70e8·39-s − 3.06e8·41-s − 1.18e8·43-s − 2.51e8·45-s − 3.44e8·47-s + 1.41e8·49-s + 1.37e9·51-s + 3.92e8·53-s − 1.30e9·55-s + 1.81e9·61-s + 1.12e9·63-s + ⋯ |
| L(s) = 1 | + 1.50·3-s − 6/5·5-s + 1.00·7-s + 1.13·9-s + 2.15·11-s − 1.25·13-s − 1.80·15-s + 2.64·17-s + 1.50·21-s + 1.62·23-s + 0.439·25-s + 1.50·27-s + 0.703·31-s + 3.24·33-s − 1.20·35-s + 1.62·37-s − 1.88·39-s − 2.64·41-s − 0.807·43-s − 1.36·45-s − 1.50·47-s + 0.500·49-s + 3.98·51-s + 0.939·53-s − 2.58·55-s + 2.14·61-s + 1.13·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(8.190861144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.190861144\) |
| \(L(6)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 6 p^{4} T + p^{10} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 122 p T + 7442 p^{2} T^{2} - 122 p^{11} T^{3} + p^{20} T^{4} \) |
| 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
−12.27548829416632039630476343896, −12.16902459606303990892942026533, −11.58853856903694357744293872828, −11.22463385870116491144280032357, −10.06981486691008358423172222584, −9.790343265996633728105400570922, −9.175832996127858949686686930990, −8.465665380127882376599884672732, −8.069005404717339546203960104211, −7.81124047283015470834013005126, −6.91055093572172385871418614281, −6.66680910871497871461114980247, −5.03467745659925098052402753579, −4.97775253539517899075786728914, −3.69656953587398310516044218991, −3.68478938256555617304236131185, −2.91619343759840951912961948598, −2.02885159368370157468766161578, −1.07724503820724280621966867026, −0.876589375462448776890139757244,
0.876589375462448776890139757244, 1.07724503820724280621966867026, 2.02885159368370157468766161578, 2.91619343759840951912961948598, 3.68478938256555617304236131185, 3.69656953587398310516044218991, 4.97775253539517899075786728914, 5.03467745659925098052402753579, 6.66680910871497871461114980247, 6.91055093572172385871418614281, 7.81124047283015470834013005126, 8.069005404717339546203960104211, 8.465665380127882376599884672732, 9.175832996127858949686686930990, 9.790343265996633728105400570922, 10.06981486691008358423172222584, 11.22463385870116491144280032357, 11.58853856903694357744293872828, 12.16902459606303990892942026533, 12.27548829416632039630476343896
