| L(s) = 1 | − 4·2-s + 6·4-s + 4·9-s + 32·11-s − 15·16-s − 16·18-s − 128·22-s − 8·23-s − 8·25-s − 20·29-s + 24·32-s + 24·36-s + 8·37-s − 20·43-s + 192·44-s + 32·46-s + 32·50-s − 4·53-s + 80·58-s − 6·64-s + 12·67-s − 48·71-s − 32·74-s − 8·79-s + 9·81-s + 80·86-s − 48·92-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s + 4/3·9-s + 9.64·11-s − 3.75·16-s − 3.77·18-s − 27.2·22-s − 1.66·23-s − 8/5·25-s − 3.71·29-s + 4.24·32-s + 4·36-s + 1.31·37-s − 3.04·43-s + 28.9·44-s + 4.71·46-s + 4.52·50-s − 0.549·53-s + 10.5·58-s − 3/4·64-s + 1.46·67-s − 5.69·71-s − 3.71·74-s − 0.900·79-s + 81-s + 8.62·86-s − 5.00·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2464579174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2464579174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 4 T^{2} + 39 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 13 | \( ( 1 - 8 T^{2} - 105 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 32 T^{2} + 735 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 36 T^{2} + 625 T^{4} + 1836 T^{6} - 147936 T^{8} + 1836 p^{2} T^{10} + 625 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 10 T + 32 T^{2} + 100 T^{3} + 883 T^{4} + 100 p T^{5} + 32 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 32 T^{2} + 63 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T - 2 T^{2} + 224 T^{3} - 1637 T^{4} + 224 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 50 T^{2} + 819 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 10 T + 4 T^{2} + 100 T^{3} + 3067 T^{4} + 100 p T^{5} + 4 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 124 T^{2} + 7354 T^{4} - 446896 T^{6} + 25507219 T^{8} - 446896 p^{2} T^{10} + 7354 p^{4} T^{12} - 124 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 2 T + 32 T^{2} - 268 T^{3} - 2237 T^{4} - 268 p T^{5} + 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 160 T^{2} + 13198 T^{4} - 870400 T^{6} + 52578643 T^{8} - 870400 p^{2} T^{10} + 13198 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 204 T^{2} + 24145 T^{4} - 2045916 T^{6} + 136536864 T^{8} - 2045916 p^{2} T^{10} + 24145 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 6 T - 92 T^{2} + 36 T^{3} + 9483 T^{4} + 36 p T^{5} - 92 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 12 T + 163 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 - 36 T^{2} + 2074 T^{4} + 411696 T^{6} - 34699341 T^{8} + 411696 p^{2} T^{10} + 2074 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 4 T - 131 T^{2} - 44 T^{3} + 14104 T^{4} - 44 p T^{5} - 131 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 256 T^{2} + 36334 T^{4} - 3948544 T^{6} + 353820979 T^{8} - 3948544 p^{2} T^{10} + 36334 p^{4} T^{12} - 256 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 128 T^{2} + 8463 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 156 T^{2} + 1594 T^{4} + 612144 T^{6} + 199691859 T^{8} + 612144 p^{2} T^{10} + 1594 p^{4} T^{12} + 156 p^{6} T^{14} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.20300532049496670503878727022, −4.12323345026879454324417362310, −4.09272948677993106939399879464, −4.03596689324754488845585026806, −3.86756356413446943214650058088, −3.82045541224050016061922398574, −3.77103208620011182823227170889, −3.52734007846086580031631707292, −3.34706615725959912474641895213, −3.20658412216382463571412602380, −3.07900589323139212756668721694, −2.97642068148274468363441712301, −2.42163862649950695287791990562, −2.19326927421326273658186290299, −2.04617374919499320029567213018, −2.02766874732862311174104165334, −1.63047515884024271954542448565, −1.53144278066200501855189139246, −1.50475030009961052893842945656, −1.48454624997801608509516176266, −1.29171414709637621478612302558, −1.23329882617812832486086057251, −1.03278302981355986740756528655, −0.55321344988203623834680773587, −0.10597068095893563475186739284,
0.10597068095893563475186739284, 0.55321344988203623834680773587, 1.03278302981355986740756528655, 1.23329882617812832486086057251, 1.29171414709637621478612302558, 1.48454624997801608509516176266, 1.50475030009961052893842945656, 1.53144278066200501855189139246, 1.63047515884024271954542448565, 2.02766874732862311174104165334, 2.04617374919499320029567213018, 2.19326927421326273658186290299, 2.42163862649950695287791990562, 2.97642068148274468363441712301, 3.07900589323139212756668721694, 3.20658412216382463571412602380, 3.34706615725959912474641895213, 3.52734007846086580031631707292, 3.77103208620011182823227170889, 3.82045541224050016061922398574, 3.86756356413446943214650058088, 4.03596689324754488845585026806, 4.09272948677993106939399879464, 4.12323345026879454324417362310, 4.20300532049496670503878727022
Plot not available for L-functions of degree greater than 10.
