| L(s) = 1 | − 8·5-s + 24·13-s + 32·25-s + 8·29-s − 24·37-s + 32·49-s − 40·53-s − 24·61-s − 192·65-s + 64·97-s + 40·101-s + 8·109-s + 144·113-s − 72·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 288·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 3.57·5-s + 6.65·13-s + 32/5·25-s + 1.48·29-s − 3.94·37-s + 32/7·49-s − 5.49·53-s − 3.07·61-s − 23.8·65-s + 6.49·97-s + 3.98·101-s + 0.766·109-s + 13.5·113-s − 6.43·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 22.1·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{64} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.668062486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.668062486\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + 4 T + 8 T^{2} + 4 T^{3} - 14 T^{4} + 4 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( 1 - 28 T^{4} + 1830 T^{8} - 28 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( 1 + 292 T^{4} - 25242 T^{8} + 292 p^{4} T^{12} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 4 T + 8 T^{2} + 92 T^{3} - 1646 T^{4} + 92 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + 12 T + 72 T^{2} + 372 T^{3} + 1886 T^{4} + 372 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 12 T^{2} + 326 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 1778 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 20 T + 200 T^{2} + 1940 T^{3} + 16882 T^{4} + 1940 p T^{5} + 200 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 1486 T^{4} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 12 T + 72 T^{2} + 660 T^{3} + 6014 T^{4} + 660 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 7588 T^{4} + 30907110 T^{8} + 7588 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 20 T^{2} - 2106 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 152 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 17828 T^{4} + 157096038 T^{8} + 17828 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 252 T^{2} + 30950 T^{4} - 252 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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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
−3.58360420534588175048098224362, −3.53914838496683116228048025740, −3.44834332934403726828827088599, −3.44410221861264077576044253812, −3.38059321100844721409057697927, −3.36393681319599717836834169897, −3.28217278261362835733137806788, −3.22030272657091140755112100217, −2.92540299877419486611044125122, −2.90189257038087699060938300773, −2.38887308464975550689726032260, −2.36285064177276608801447804513, −2.33134873135441393669147386808, −2.04472446790216041952876306353, −1.74192111962352485336687987767, −1.69210995717753153390305351053, −1.66998164780330281319300840538, −1.63029110280207490813326234871, −1.25146934833153941007909044998, −1.08731063889929318421155516574, −0.902642457026024816345590562384, −0.75771232711785538545055598942, −0.68731883576978233903983795543, −0.37768616686863925024896562651, −0.37531651768677810398260756793,
0.37531651768677810398260756793, 0.37768616686863925024896562651, 0.68731883576978233903983795543, 0.75771232711785538545055598942, 0.902642457026024816345590562384, 1.08731063889929318421155516574, 1.25146934833153941007909044998, 1.63029110280207490813326234871, 1.66998164780330281319300840538, 1.69210995717753153390305351053, 1.74192111962352485336687987767, 2.04472446790216041952876306353, 2.33134873135441393669147386808, 2.36285064177276608801447804513, 2.38887308464975550689726032260, 2.90189257038087699060938300773, 2.92540299877419486611044125122, 3.22030272657091140755112100217, 3.28217278261362835733137806788, 3.36393681319599717836834169897, 3.38059321100844721409057697927, 3.44410221861264077576044253812, 3.44834332934403726828827088599, 3.53914838496683116228048025740, 3.58360420534588175048098224362
Plot not available for L-functions of degree greater than 10.
