| L(s) = 1 | + 32·2-s − 366·3-s + 512·4-s − 3.75e3·5-s − 1.17e4·6-s − 1.68e4·7-s + 6.69e4·9-s − 1.20e5·10-s − 3.46e5·11-s − 1.87e5·12-s − 4.65e5·13-s − 5.38e5·14-s + 1.37e6·15-s − 2.62e5·16-s + 3.76e6·17-s + 2.14e6·18-s − 1.92e6·20-s + 6.15e6·21-s − 1.10e7·22-s − 1.04e7·23-s + 4.29e6·25-s − 1.48e7·26-s − 2.16e7·27-s − 8.60e6·28-s + 4.39e7·30-s − 2.01e7·31-s − 8.38e6·32-s + ⋯ |
| L(s) = 1 | + 2-s − 1.50·3-s + 1/2·4-s − 6/5·5-s − 1.50·6-s − 1.00·7-s + 1.13·9-s − 6/5·10-s − 2.15·11-s − 0.753·12-s − 1.25·13-s − 1.00·14-s + 1.80·15-s − 1/4·16-s + 2.64·17-s + 1.13·18-s − 3/5·20-s + 1.50·21-s − 2.15·22-s − 1.62·23-s + 0.439·25-s − 1.25·26-s − 1.50·27-s − 0.500·28-s + 1.80·30-s − 0.703·31-s − 1/4·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.1769415757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1769415757\) |
| \(L(6)\) |
|
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 | $C_2$ | \( 1 - p^{5} T + p^{9} T^{2} \) |
| 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
−19.35384534432907741148630021168, −18.23498356647136324701241640361, −17.09951454350802975732890847444, −16.38129556966235166425389175243, −16.06842489748991662543680974712, −15.31579883164481178863418187891, −14.55785196726634088092852042834, −13.29615959638143144285255479439, −12.69033673523140136186095112993, −11.86043780059560216164986306203, −11.83386948381122037982389703625, −10.26104920901986566020196732926, −10.08791032879372778294701243055, −7.82031297319141331675296873376, −7.35077607274388347352801449420, −5.63607573809750764452176700342, −5.51890087363622651618771930643, −4.11258631128076034722736884796, −2.91353861480390921986173232870, −0.21298830104623525771142697366,
0.21298830104623525771142697366, 2.91353861480390921986173232870, 4.11258631128076034722736884796, 5.51890087363622651618771930643, 5.63607573809750764452176700342, 7.35077607274388347352801449420, 7.82031297319141331675296873376, 10.08791032879372778294701243055, 10.26104920901986566020196732926, 11.83386948381122037982389703625, 11.86043780059560216164986306203, 12.69033673523140136186095112993, 13.29615959638143144285255479439, 14.55785196726634088092852042834, 15.31579883164481178863418187891, 16.06842489748991662543680974712, 16.38129556966235166425389175243, 17.09951454350802975732890847444, 18.23498356647136324701241640361, 19.35384534432907741148630021168
