| L(s) = 1 | − 8·2-s + 2·3-s + 30·4-s − 10·5-s − 16·6-s − 80·8-s + 23·9-s + 80·10-s + 14·11-s + 60·12-s − 100·13-s − 20·15-s + 192·16-s − 50·17-s − 184·18-s + 36·19-s − 300·20-s − 112·22-s − 244·23-s − 160·24-s + 25·25-s + 800·26-s + 22·27-s − 52·29-s + 160·30-s − 120·31-s − 480·32-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.384·3-s + 15/4·4-s − 0.894·5-s − 1.08·6-s − 3.53·8-s + 0.851·9-s + 2.52·10-s + 0.383·11-s + 1.44·12-s − 2.13·13-s − 0.344·15-s + 3·16-s − 0.713·17-s − 2.40·18-s + 0.434·19-s − 3.35·20-s − 1.08·22-s − 2.21·23-s − 1.36·24-s + 1/5·25-s + 6.03·26-s + 0.156·27-s − 0.332·29-s + 0.973·30-s − 0.695·31-s − 2.65·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0003629669876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0003629669876\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{3} T + 17 p T^{2} + 7 p^{4} T^{3} + 81 p^{2} T^{4} + 7 p^{7} T^{5} + 17 p^{7} T^{6} + p^{12} T^{7} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 T - 19 T^{2} + 62 T^{3} - 308 T^{4} + 62 p^{3} T^{5} - 19 p^{6} T^{6} - 2 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 14 T - 467 T^{2} + 27986 T^{3} - 1592868 T^{4} + 27986 p^{3} T^{5} - 467 p^{6} T^{6} - 14 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 50 T + 4987 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 50 T - 4079 T^{2} - 9550 p T^{3} + 23748 p^{2} T^{4} - 9550 p^{4} T^{5} - 4079 p^{6} T^{6} + 50 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 36 T - 8874 T^{2} + 127728 T^{3} + 47493755 T^{4} + 127728 p^{3} T^{5} - 8874 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 244 T + 29566 T^{2} + 1375184 T^{3} + 25790499 T^{4} + 1375184 p^{3} T^{5} + 29566 p^{6} T^{6} + 244 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 26 T + 47795 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 120 T + 16018 T^{2} - 7344000 T^{3} - 1313876157 T^{4} - 7344000 p^{3} T^{5} + 16018 p^{6} T^{6} + 120 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 564 T + 144466 T^{2} + 40790736 T^{3} + 11469133803 T^{4} + 40790736 p^{3} T^{5} + 144466 p^{6} T^{6} + 564 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 p T + 133986 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 260 T + 166666 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 350 T - 80923 T^{2} - 1478050 T^{3} + 17883384100 T^{4} - 1478050 p^{3} T^{5} - 80923 p^{6} T^{6} + 350 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 56 T - 262634 T^{2} + 1791104 T^{3} + 48002453499 T^{4} + 1791104 p^{3} T^{5} - 262634 p^{6} T^{6} - 56 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 616 T + 174077 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 336 T - 345962 T^{2} - 1645056 T^{3} + 133405042827 T^{4} - 1645056 p^{3} T^{5} - 345962 p^{6} T^{6} - 336 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 152 T - 576006 T^{2} + 367232 T^{3} + 261525581579 T^{4} + 367232 p^{3} T^{5} - 576006 p^{6} T^{6} - 152 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 952 T + p^{3} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 676 T - 198630 T^{2} + 82761328 T^{3} + 100713568355 T^{4} + 82761328 p^{3} T^{5} - 198630 p^{6} T^{6} - 676 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1014 T - 91923 T^{2} + 135917574 T^{3} + 504638390996 T^{4} + 135917574 p^{3} T^{5} - 91923 p^{6} T^{6} + 1014 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 376 T + 458918 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 216 T - 1371074 T^{2} + 1683072 T^{3} + 1480086028275 T^{4} + 1683072 p^{3} T^{5} - 1371074 p^{6} T^{6} + 216 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 2742 T + 3608187 T^{2} + 2742 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.404602088416747718113850526790, −8.171247123008942661165983142654, −8.005918632432870636823739732500, −7.61165272321353167302387920519, −7.44545626872260355057374608984, −7.18877730305463269648800434869, −7.13890115981328138917842738550, −6.74277021331907908320851467660, −6.47190589734275662209264548578, −6.05586929190493162373579043994, −5.71292979720318313336184966354, −5.32299636096769760632053146402, −5.16902821837246979806840328257, −4.43065916730902932818080815105, −4.37131770185746373263737372946, −4.13260576977156541378370543283, −3.67581445552852315430229999852, −3.18522772534661505679495050635, −2.82702218361181578424852030203, −2.50846951548437255664620881159, −1.86968840201210435871403332402, −1.53223308214757423398115452532, −1.44627153322457036799053257847, −0.30060263136164210245120520787, −0.01627518568284953883917883897,
0.01627518568284953883917883897, 0.30060263136164210245120520787, 1.44627153322457036799053257847, 1.53223308214757423398115452532, 1.86968840201210435871403332402, 2.50846951548437255664620881159, 2.82702218361181578424852030203, 3.18522772534661505679495050635, 3.67581445552852315430229999852, 4.13260576977156541378370543283, 4.37131770185746373263737372946, 4.43065916730902932818080815105, 5.16902821837246979806840328257, 5.32299636096769760632053146402, 5.71292979720318313336184966354, 6.05586929190493162373579043994, 6.47190589734275662209264548578, 6.74277021331907908320851467660, 7.13890115981328138917842738550, 7.18877730305463269648800434869, 7.44545626872260355057374608984, 7.61165272321353167302387920519, 8.005918632432870636823739732500, 8.171247123008942661165983142654, 8.404602088416747718113850526790
