| L(s) = 1 | − 5·2-s + 12·4-s + 3·5-s − 7-s − 20·8-s − 15·10-s − 11-s + 4·13-s + 5·14-s + 30·16-s − 6·17-s − 7·19-s + 36·20-s + 5·22-s − 22·23-s + 10·25-s − 20·26-s − 12·28-s − 6·29-s − 15·31-s − 45·32-s + 30·34-s − 3·35-s − 6·37-s + 35·38-s − 60·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 3.53·2-s + 6·4-s + 1.34·5-s − 0.377·7-s − 7.07·8-s − 4.74·10-s − 0.301·11-s + 1.10·13-s + 1.33·14-s + 15/2·16-s − 1.45·17-s − 1.60·19-s + 8.04·20-s + 1.06·22-s − 4.58·23-s + 2·25-s − 3.92·26-s − 2.26·28-s − 1.11·29-s − 2.69·31-s − 7.95·32-s + 5.14·34-s − 0.507·35-s − 0.986·37-s + 5.67·38-s − 9.48·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1678907678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1678907678\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 + 5 T + 13 T^{2} + 25 T^{3} + 39 T^{4} + 25 p T^{5} + 13 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 3 T - T^{2} + 3 T^{3} + 16 T^{4} + 3 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 6 T + 19 T^{2} + 132 T^{3} + 829 T^{4} + 132 p T^{5} + 19 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 7 T + 15 T^{2} + 107 T^{3} + 824 T^{4} + 107 p T^{5} + 15 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 11 T + 75 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 132 T^{3} - 995 T^{4} - 132 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 15 T + 69 T^{2} + 95 T^{3} + 36 T^{4} + 95 p T^{5} + 69 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 6 T - 21 T^{2} - 248 T^{3} - 471 T^{4} - 248 p T^{5} - 21 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 6 T + 35 T^{2} - 84 T^{3} - 371 T^{4} - 84 p T^{5} + 35 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_4\times C_2$ | \( 1 - 10 T + 13 T^{2} - 200 T^{3} + 3549 T^{4} - 200 p T^{5} + 13 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 14 T + 43 T^{2} + 650 T^{3} - 8799 T^{4} + 650 p T^{5} + 43 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 6 T - 43 T^{2} + 102 T^{3} + 3025 T^{4} + 102 p T^{5} - 43 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 24 T + 185 T^{2} - 456 T^{3} + 49 T^{4} - 456 p T^{5} + 185 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 22 T + 111 T^{2} - 1054 T^{3} - 17431 T^{4} - 1054 p T^{5} + 111 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 10 T + 21 T^{2} + 580 T^{3} - 7459 T^{4} + 580 p T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 2 T + 121 T^{2} - 736 T^{3} + 7989 T^{4} - 736 p T^{5} + 121 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 9 T + 197 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_4\times C_2$ | \( 1 - 6 T - 61 T^{2} + 948 T^{3} + 229 T^{4} + 948 p T^{5} - 61 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.72283418653168875028480814210, −7.46806371381200743035634387499, −7.02946013813497735730244987672, −7.02737752621503957268480407873, −6.90174671999689908626027482301, −6.23066863500729309798997485019, −6.16868064073220007114078794748, −6.03209973110451030084505500144, −5.99177639184461849666941453355, −5.57114258060562523641974237498, −5.39207939116513542132151188738, −5.10506057260647229504080679984, −4.49749686978425586062707623859, −4.17139758756295600897947975613, −3.77583285364392106280796744713, −3.76784620495717196351042213406, −3.73476078243174756628525878601, −2.79409494241422522582195023145, −2.39201394884247906088713761290, −2.23324114731280969912239438286, −2.18883715489788826614869612617, −1.57314477923992128282066182066, −1.53523579922300743074563566764, −0.46585796852888750620517922403, −0.43763971164208055440163190000,
0.43763971164208055440163190000, 0.46585796852888750620517922403, 1.53523579922300743074563566764, 1.57314477923992128282066182066, 2.18883715489788826614869612617, 2.23324114731280969912239438286, 2.39201394884247906088713761290, 2.79409494241422522582195023145, 3.73476078243174756628525878601, 3.76784620495717196351042213406, 3.77583285364392106280796744713, 4.17139758756295600897947975613, 4.49749686978425586062707623859, 5.10506057260647229504080679984, 5.39207939116513542132151188738, 5.57114258060562523641974237498, 5.99177639184461849666941453355, 6.03209973110451030084505500144, 6.16868064073220007114078794748, 6.23066863500729309798997485019, 6.90174671999689908626027482301, 7.02737752621503957268480407873, 7.02946013813497735730244987672, 7.46806371381200743035634387499, 7.72283418653168875028480814210
