| L(s) = 1 | + 24·9-s − 32·13-s + 8·17-s − 120·29-s + 48·37-s + 184·49-s − 160·53-s + 192·61-s + 104·73-s + 290·81-s + 328·89-s − 104·97-s + 56·101-s − 288·109-s + 168·113-s − 768·117-s + 116·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 192·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 2.46·13-s + 8/17·17-s − 4.13·29-s + 1.29·37-s + 3.75·49-s − 3.01·53-s + 3.14·61-s + 1.42·73-s + 3.58·81-s + 3.68·89-s − 1.07·97-s + 0.554·101-s − 2.64·109-s + 1.48·113-s − 6.56·117-s + 0.958·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.25·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(5.828561591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.828561591\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 8 p T^{2} + 286 T^{4} - 8 p^{5} T^{6} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 13246 T^{4} - 184 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - 116 T^{2} + 246 p^{2} T^{4} - 116 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 16 T + 222 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 502 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 516 T^{2} + 142886 T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1304 T^{2} + 983166 T^{4} - 1304 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 60 T + 2502 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2004 T^{2} + 2779046 T^{4} - 2004 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 24 T + 1902 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 1064 T^{2} + 7036126 T^{4} + 1064 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4344 T^{2} + 11292926 T^{4} - 4344 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 80 T + 3838 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1732 T^{2} - 7783322 T^{4} - 1732 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 96 T + 7326 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12888 T^{2} + 75532958 T^{4} - 12888 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 11476 T^{2} + 76060006 T^{4} - 11476 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 52 T + 9334 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8132 T^{2} + 44251398 T^{4} - 8132 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 14984 T^{2} + 130442206 T^{4} + 14984 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 164 T + 21286 T^{2} - 164 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 52 T + 7974 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.66202671979479310556884583535, −6.40968309733624713147878196115, −5.92642417251166308177642838285, −5.84201537482938302019448069413, −5.69343106830526743283734373782, −5.28966597144411211094335500783, −5.24010900026741598484695518750, −4.94769028702021867139116151917, −4.81295465799411542582976391645, −4.39879913194923394790118393031, −4.36655261449435464226593884500, −4.01627041224345091503763147149, −3.92402788561924188931405898966, −3.56310390750722921511211139109, −3.51790446346761077785981258398, −3.11356795880908483674537700068, −2.66629476183954394957977774971, −2.36965944801425975810875218138, −2.11262390725500746739340652327, −2.06927231113832866645406226777, −1.73990089656100701867212961956, −1.36928438562084695227203948095, −0.965225536657815427560793885469, −0.52428188733126016509410861479, −0.40571131558124061521968746157,
0.40571131558124061521968746157, 0.52428188733126016509410861479, 0.965225536657815427560793885469, 1.36928438562084695227203948095, 1.73990089656100701867212961956, 2.06927231113832866645406226777, 2.11262390725500746739340652327, 2.36965944801425975810875218138, 2.66629476183954394957977774971, 3.11356795880908483674537700068, 3.51790446346761077785981258398, 3.56310390750722921511211139109, 3.92402788561924188931405898966, 4.01627041224345091503763147149, 4.36655261449435464226593884500, 4.39879913194923394790118393031, 4.81295465799411542582976391645, 4.94769028702021867139116151917, 5.24010900026741598484695518750, 5.28966597144411211094335500783, 5.69343106830526743283734373782, 5.84201537482938302019448069413, 5.92642417251166308177642838285, 6.40968309733624713147878196115, 6.66202671979479310556884583535
