| L(s) = 1 | − 8·19-s + 24·31-s − 16·37-s − 24·43-s + 24·49-s − 16·61-s + 32·67-s − 56·79-s + 32·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 1.83·19-s + 4.31·31-s − 2.63·37-s − 3.65·43-s + 24/7·49-s − 2.04·61-s + 3.90·67-s − 6.30·79-s + 3.06·109-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4315398933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4315398933\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 12 T^{4} - 506 T^{8} - 12 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 12 T^{2} + 106 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 60 T^{4} - 14618 T^{8} - 60 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 194 T^{4} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 28 T^{2} + 662 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 4 T + 8 T^{2} + 28 T^{3} - 46 T^{4} + 28 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 12 T^{2} + 646 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 + 180 T^{4} - 772538 T^{8} + 180 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( ( 1 - 6 T + 64 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 8 T + 32 T^{2} + 248 T^{3} + 1886 T^{4} + 248 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 60 T^{2} + 4150 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 12 T + 72 T^{2} + 564 T^{3} + 4402 T^{4} + 564 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 20 T^{2} + 4070 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 5772 T^{4} + 23807110 T^{8} - 5772 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( 1 - 7452 T^{4} + 27767206 T^{8} - 7452 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 8 T + 32 T^{2} + 440 T^{3} + 6014 T^{4} + 440 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 92 T^{2} + 5030 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 228 T^{2} + 23206 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 14 T + 200 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 20100 T^{4} + 185294374 T^{8} + 20100 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 260 T^{2} + 30950 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{4} \) |
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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
−4.27891601664286915716661209791, −4.23291424479381722408842449064, −4.04120942442293586217752641494, −3.80037909360911735469316046725, −3.50169875901892915645325533870, −3.40917565127346452779587533444, −3.38838930548990151396166292520, −3.37727738513199372264794212858, −3.16628915083695519403341980304, −2.99532283288737381141027860437, −2.79109126198268768784270056265, −2.72232086399079977449194396130, −2.64856487520332759208697655957, −2.39976235403248726819098093211, −2.09676202345228768698943837084, −2.03179737634615792712051471915, −1.97658663488879828517040777150, −1.87694649151343520713365953277, −1.68223547966626059777731504254, −1.36294851104108039864558930082, −1.10838198232179727568753270205, −0.927120962253340371606931743162, −0.917076945509328655634521290283, −0.44289040598309111264361564255, −0.084090252463627193104924800049,
0.084090252463627193104924800049, 0.44289040598309111264361564255, 0.917076945509328655634521290283, 0.927120962253340371606931743162, 1.10838198232179727568753270205, 1.36294851104108039864558930082, 1.68223547966626059777731504254, 1.87694649151343520713365953277, 1.97658663488879828517040777150, 2.03179737634615792712051471915, 2.09676202345228768698943837084, 2.39976235403248726819098093211, 2.64856487520332759208697655957, 2.72232086399079977449194396130, 2.79109126198268768784270056265, 2.99532283288737381141027860437, 3.16628915083695519403341980304, 3.37727738513199372264794212858, 3.38838930548990151396166292520, 3.40917565127346452779587533444, 3.50169875901892915645325533870, 3.80037909360911735469316046725, 4.04120942442293586217752641494, 4.23291424479381722408842449064, 4.27891601664286915716661209791
Plot not available for L-functions of degree greater than 10.
