| L(s) = 1 | + 4·4-s − 8·7-s + 64·13-s + 4·16-s + 112·19-s − 64·25-s − 32·28-s − 32·31-s − 152·37-s + 88·43-s + 4·49-s + 256·52-s + 52·61-s − 16·64-s + 40·67-s + 448·73-s + 448·76-s − 104·79-s − 512·91-s − 32·97-s − 256·100-s + 112·103-s − 128·109-s − 32·112-s − 52·121-s − 128·124-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 8/7·7-s + 4.92·13-s + 1/4·16-s + 5.89·19-s − 2.55·25-s − 8/7·28-s − 1.03·31-s − 4.10·37-s + 2.04·43-s + 4/49·49-s + 4.92·52-s + 0.852·61-s − 1/4·64-s + 0.597·67-s + 6.13·73-s + 5.89·76-s − 1.31·79-s − 5.62·91-s − 0.329·97-s − 2.55·100-s + 1.08·103-s − 1.17·109-s − 2/7·112-s − 0.429·121-s − 1.03·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.955567540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.955567540\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 64 T^{2} + 413 p T^{4} + 49984 T^{6} + 1174336 T^{8} + 49984 p^{4} T^{10} + 413 p^{9} T^{12} + 64 p^{12} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 + 4 T + 22 T^{2} - 416 T^{3} - 3149 T^{4} - 416 p^{2} T^{5} + 22 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 26 T^{2} - 13965 T^{4} + 26 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 - 32 T + 457 T^{2} - 7328 T^{3} + 119872 T^{4} - 7328 p^{2} T^{5} + 457 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 - 256 T^{2} + 31551 T^{4} - 256 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 19 | \( ( 1 - 28 T + 810 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 23 | \( ( 1 + 842 T^{2} + 429123 T^{4} + 842 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( 1 + 1312 T^{2} - 68879 T^{4} + 492867232 T^{6} + 1487178071104 T^{8} + 492867232 p^{4} T^{10} - 68879 p^{8} T^{12} + 1312 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 + 8 T - 897 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 + 38 T + 1371 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 41 | \( 1 + 4960 T^{2} + 12861886 T^{4} + 30197432320 T^{6} + 60317717755075 T^{8} + 30197432320 p^{4} T^{10} + 12861886 p^{8} T^{12} + 4960 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 - 44 T - 1274 T^{2} + 21472 T^{3} + 3305635 T^{4} + 21472 p^{2} T^{5} - 1274 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 + 4130 T^{2} + 12177219 T^{4} + 4130 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 - 3424 T^{2} + 8198754 T^{4} - 3424 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 59 | \( 1 + 1828 T^{2} - 16689686 T^{4} - 7683910256 T^{6} + 213930160568755 T^{8} - 7683910256 p^{4} T^{10} - 16689686 p^{8} T^{12} + 1828 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 - 13 T - 3552 T^{2} - 13 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 20 T - 7706 T^{2} + 17440 T^{3} + 43760515 T^{4} + 17440 p^{2} T^{5} - 7706 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 9652 T^{2} + 49230438 T^{4} - 9652 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 112 T + 13551 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 + 52 T - 5162 T^{2} - 240032 T^{3} + 6048211 T^{4} - 240032 p^{2} T^{5} - 5162 p^{4} T^{6} + 52 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 + 16612 T^{2} + 139485994 T^{4} + 690326743696 T^{6} + 3275935507518835 T^{8} + 690326743696 p^{4} T^{10} + 139485994 p^{8} T^{12} + 16612 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 7456 T^{2} + 104845119 T^{4} - 7456 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T - 11714 T^{2} - 109568 T^{3} + 52342915 T^{4} - 109568 p^{2} T^{5} - 11714 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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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
−5.60485542582848396454439465776, −5.53999541989370817658652714399, −5.36960413692600506597861872516, −5.32812207025694996889567957564, −5.11124413964052008100214576590, −5.06354840834366759172966862329, −4.64428810072702964938002112695, −4.45808319919625303861505557278, −4.07204994267605133529850240299, −3.89760320827542638353425258702, −3.74682708853466206035149213783, −3.52404467737641084309733127247, −3.50381770609176535938664533553, −3.43765566328830062737882266674, −3.41545032922451652848766076344, −3.13177559173440555740712333541, −2.86360404950262065881847506330, −2.49464139978026534722499683194, −2.08589254410539527013732739462, −2.02521181221762547411533141532, −1.68394490542215810343124684666, −1.33113292888396823755592074062, −1.05915895591329525924221133437, −1.01435283290995646115564892077, −0.50119531386977477575680577539,
0.50119531386977477575680577539, 1.01435283290995646115564892077, 1.05915895591329525924221133437, 1.33113292888396823755592074062, 1.68394490542215810343124684666, 2.02521181221762547411533141532, 2.08589254410539527013732739462, 2.49464139978026534722499683194, 2.86360404950262065881847506330, 3.13177559173440555740712333541, 3.41545032922451652848766076344, 3.43765566328830062737882266674, 3.50381770609176535938664533553, 3.52404467737641084309733127247, 3.74682708853466206035149213783, 3.89760320827542638353425258702, 4.07204994267605133529850240299, 4.45808319919625303861505557278, 4.64428810072702964938002112695, 5.06354840834366759172966862329, 5.11124413964052008100214576590, 5.32812207025694996889567957564, 5.36960413692600506597861872516, 5.53999541989370817658652714399, 5.60485542582848396454439465776
Plot not available for L-functions of degree greater than 10.
