| L(s) = 1 | − 8·2-s + 40·4-s + 2·7-s − 146·8-s − 2·9-s − 16·14-s + 434·16-s + 16·18-s − 2·19-s − 20·25-s + 80·28-s − 1.09e3·32-s − 80·36-s + 16·38-s + 14·41-s + 36·47-s + 15·49-s + 160·50-s − 292·56-s − 46·59-s − 4·63-s + 2.41e3·64-s + 48·67-s − 26·71-s + 292·72-s − 80·76-s − 11·81-s + ⋯ |
| L(s) = 1 | − 5.65·2-s + 20·4-s + 0.755·7-s − 51.6·8-s − 2/3·9-s − 4.27·14-s + 108.5·16-s + 3.77·18-s − 0.458·19-s − 4·25-s + 15.1·28-s − 193.·32-s − 13.3·36-s + 2.59·38-s + 2.18·41-s + 5.25·47-s + 15/7·49-s + 22.6·50-s − 39.0·56-s − 5.98·59-s − 0.503·63-s + 301.·64-s + 5.86·67-s − 3.08·71-s + 34.4·72-s − 9.17·76-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3651700638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3651700638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( ( 1 + p^{2} T + p^{2} T^{2} - 7 T^{3} - 21 T^{4} - 7 p T^{5} + p^{4} T^{6} + p^{5} T^{7} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 + 2 T^{2} + 5 p T^{4} + 22 T^{6} + 169 T^{8} + 22 p^{2} T^{10} + 5 p^{5} T^{12} + 2 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 13 p T^{5} - 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( 1 - 4 T^{2} - 105 T^{4} + 904 T^{6} + 9089 T^{8} + 904 p^{2} T^{10} - 105 p^{4} T^{12} - 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 + 46 T^{2} + 1047 T^{4} + 17438 T^{6} + 244505 T^{8} + 17438 p^{2} T^{10} + 1047 p^{4} T^{12} + 46 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( 1 - 12 T^{2} + 575 T^{4} - 13122 T^{6} + 166429 T^{8} - 13122 p^{2} T^{10} + 575 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( ( 1 + T - 18 T^{2} - 37 T^{3} + 305 T^{4} - 37 p T^{5} - 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 64 T^{2} + 1047 T^{4} + 18958 T^{6} - 981475 T^{8} + 18958 p^{2} T^{10} + 1047 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( 1 - 36 T^{2} + 1895 T^{4} - 82914 T^{6} + 2002669 T^{8} - 82914 p^{2} T^{10} + 1895 p^{4} T^{12} - 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 7 T + 28 T^{2} - 389 T^{3} + 3975 T^{4} - 389 p T^{5} + 28 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 144 T^{2} + 6287 T^{4} + 99078 T^{6} - 15616595 T^{8} + 99078 p^{2} T^{10} + 6287 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 - 18 T + 97 T^{2} - 150 T^{3} + 121 T^{4} - 150 p T^{5} + 97 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 + 182 T^{2} + 16635 T^{4} + 1141492 T^{6} + 65975189 T^{8} + 1141492 p^{2} T^{10} + 16635 p^{4} T^{12} + 182 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 + 23 T + 190 T^{2} + 853 T^{3} + 4569 T^{4} + 853 p T^{5} + 190 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 30 T^{2} + 62 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 6 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 + 13 T - 2 T^{2} - 349 T^{3} + 405 T^{4} - 349 p T^{5} - 2 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 - 128 T^{2} + 11055 T^{4} - 732928 T^{6} + 34902689 T^{8} - 732928 p^{2} T^{10} + 11055 p^{4} T^{12} - 128 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 - 230 T^{2} + 13719 T^{4} + 1400990 T^{6} - 238413799 T^{8} + 1400990 p^{2} T^{10} + 13719 p^{4} T^{12} - 230 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 156 T^{2} + 17447 T^{4} - 1647048 T^{6} + 136747105 T^{8} - 1647048 p^{2} T^{10} + 17447 p^{4} T^{12} - 156 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( 1 + 182 T^{2} + 24303 T^{4} + 2899174 T^{6} + 320768105 T^{8} + 2899174 p^{2} T^{10} + 24303 p^{4} T^{12} + 182 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 7 T - 48 T^{2} + 1015 T^{3} - 2449 T^{4} + 1015 p T^{5} - 48 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.24662154656298983912197535712, −4.23462203316233281286054875654, −3.88234715899370298186577880418, −3.84218184714597242439284834567, −3.80259923344531833903155818156, −3.34012573842647141707959221162, −3.31307937646615009752001680709, −3.16816986618836336880660689836, −2.98759984783902859011410891607, −2.87573278740722618099513233777, −2.62113081443131049232458024983, −2.56498900347024750073570196202, −2.26413670676650109580625984147, −2.14505521880063354773859126505, −2.09379080046796416555893739018, −2.06647689506935389110681968022, −2.05357828809151524236326650463, −1.79169848815250211564814837005, −1.48274138782736873737119752522, −1.41539560312944308309697385516, −1.18325877995981000119090038211, −0.931774402547359074935593346477, −0.75386575426038847767837004900, −0.70828123505456031168743431621, −0.17935042567853505727369995285,
0.17935042567853505727369995285, 0.70828123505456031168743431621, 0.75386575426038847767837004900, 0.931774402547359074935593346477, 1.18325877995981000119090038211, 1.41539560312944308309697385516, 1.48274138782736873737119752522, 1.79169848815250211564814837005, 2.05357828809151524236326650463, 2.06647689506935389110681968022, 2.09379080046796416555893739018, 2.14505521880063354773859126505, 2.26413670676650109580625984147, 2.56498900347024750073570196202, 2.62113081443131049232458024983, 2.87573278740722618099513233777, 2.98759984783902859011410891607, 3.16816986618836336880660689836, 3.31307937646615009752001680709, 3.34012573842647141707959221162, 3.80259923344531833903155818156, 3.84218184714597242439284834567, 3.88234715899370298186577880418, 4.23462203316233281286054875654, 4.24662154656298983912197535712
Plot not available for L-functions of degree greater than 10.
