| L(s) = 1 | − 8·7-s − 8·17-s + 8·19-s − 8·23-s − 16·37-s − 8·41-s + 8·47-s + 32·49-s + 16·53-s − 24·59-s + 24·61-s − 16·67-s + 8·79-s − 81-s − 16·83-s − 16·97-s − 16·101-s + 24·103-s + 16·107-s − 8·113-s + 64·119-s + 20·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 1.94·17-s + 1.83·19-s − 1.66·23-s − 2.63·37-s − 1.24·41-s + 1.16·47-s + 32/7·49-s + 2.19·53-s − 3.12·59-s + 3.07·61-s − 1.95·67-s + 0.900·79-s − 1/9·81-s − 1.75·83-s − 1.62·97-s − 1.59·101-s + 2.36·103-s + 1.54·107-s − 0.752·113-s + 5.86·119-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4506141733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4506141733\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 88 T^{3} + 226 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 72 T^{3} + 98 T^{4} + 72 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1040 T^{3} + 7666 T^{4} + 1040 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 312 T^{3} + 2978 T^{4} - 312 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1104 T^{3} + 9266 T^{4} - 1104 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1328 T^{3} + 13522 T^{4} + 1328 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 - 10654 T^{4} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1584 T^{3} + 19346 T^{4} + 1584 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 220 T^{2} + 23334 T^{4} - 220 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 1808 T^{3} + 25282 T^{4} + 1808 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.28962324618692500328201040018, −6.22812807172438260385761501736, −6.08212275322934729756707377655, −5.77649532351273367347347993315, −5.60798983700397566341162729350, −5.39720317352598405899581227668, −5.14236232417400918361305704794, −4.98310049346535623900339946665, −4.62403750125294272729464511705, −4.45461201678199027693791811386, −4.07856458061355365539240345666, −4.00976553435193722746872115078, −3.64299747293296605645983426916, −3.49145947931950027212592921044, −3.44878879749905072485842671809, −3.04280460282129004242248225728, −3.01068720426165556213122210673, −2.39913310276690965537985683980, −2.37454914260167994620165705581, −2.29354309161188227348279134076, −1.76606513770863959375149538168, −1.26336327674951275245455379636, −1.20999218572391177607886582203, −0.30592702913386243815752415488, −0.26820000552855567427111166051,
0.26820000552855567427111166051, 0.30592702913386243815752415488, 1.20999218572391177607886582203, 1.26336327674951275245455379636, 1.76606513770863959375149538168, 2.29354309161188227348279134076, 2.37454914260167994620165705581, 2.39913310276690965537985683980, 3.01068720426165556213122210673, 3.04280460282129004242248225728, 3.44878879749905072485842671809, 3.49145947931950027212592921044, 3.64299747293296605645983426916, 4.00976553435193722746872115078, 4.07856458061355365539240345666, 4.45461201678199027693791811386, 4.62403750125294272729464511705, 4.98310049346535623900339946665, 5.14236232417400918361305704794, 5.39720317352598405899581227668, 5.60798983700397566341162729350, 5.77649532351273367347347993315, 6.08212275322934729756707377655, 6.22812807172438260385761501736, 6.28962324618692500328201040018
