| L(s) = 1 | + 3-s − 6·5-s − 2·7-s + 9-s + 3·11-s + 10·13-s − 6·15-s − 6·17-s − 19-s − 2·21-s − 12·23-s + 13·25-s − 4·27-s + 9·29-s − 4·31-s + 3·33-s + 12·35-s + 8·37-s + 10·39-s − 6·41-s + 5·43-s − 6·45-s − 12·47-s + 49-s − 6·51-s − 24·53-s − 18·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2.68·5-s − 0.755·7-s + 1/3·9-s + 0.904·11-s + 2.77·13-s − 1.54·15-s − 1.45·17-s − 0.229·19-s − 0.436·21-s − 2.50·23-s + 13/5·25-s − 0.769·27-s + 1.67·29-s − 0.718·31-s + 0.522·33-s + 2.02·35-s + 1.31·37-s + 1.60·39-s − 0.937·41-s + 0.762·43-s − 0.894·45-s − 1.75·47-s + 1/7·49-s − 0.840·51-s − 3.29·53-s − 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5669861347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5669861347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - T + 5 T^{3} - 11 T^{4} + 5 p T^{5} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 10 T^{2} + 9 T^{3} + 141 T^{4} + 9 p T^{5} - 10 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + T + 10 T^{2} - 47 T^{3} - 299 T^{4} - 47 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 6 T - 34 T^{2} - 72 T^{3} + 1743 T^{4} - 72 p T^{5} - 34 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 5 T - 20 T^{2} + 205 T^{3} - 899 T^{4} + 205 p T^{5} - 20 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 121 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 94 T^{2} - 72 T^{3} + 9303 T^{4} - 72 p T^{5} - 94 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 + 3 T - 124 T^{2} - 135 T^{3} + 8967 T^{4} - 135 p T^{5} - 124 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 23 T + 250 T^{2} - 1955 T^{3} + 17119 T^{4} - 1955 p T^{5} + 250 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.389949470308854772880028733866, −7.958607983584704688710525782500, −7.921293429880098905389763300372, −7.74027022823604518844485983081, −7.52071530922316272284849211141, −6.86211201622641231065087748442, −6.74339628268830533111895161162, −6.50034053240271318757754577619, −6.36788658552930167537781036919, −6.32354088944258437282584434203, −5.66077331024911153162038227874, −5.63200636284629264049814443990, −5.02040077863399731901928291529, −4.51295236775985306777952248014, −4.47224557371159307375186829560, −4.08097707525387481797908406776, −3.77686019569163612792330940553, −3.72126341657009647447545791621, −3.56368384545481240758219465395, −3.33785266248429629243502448031, −2.55643446952333234602958275759, −2.35689805768859089583005864978, −1.65732336397060433724793382099, −1.26583832796705707694123099848, −0.31962908619490817082430572195,
0.31962908619490817082430572195, 1.26583832796705707694123099848, 1.65732336397060433724793382099, 2.35689805768859089583005864978, 2.55643446952333234602958275759, 3.33785266248429629243502448031, 3.56368384545481240758219465395, 3.72126341657009647447545791621, 3.77686019569163612792330940553, 4.08097707525387481797908406776, 4.47224557371159307375186829560, 4.51295236775985306777952248014, 5.02040077863399731901928291529, 5.63200636284629264049814443990, 5.66077331024911153162038227874, 6.32354088944258437282584434203, 6.36788658552930167537781036919, 6.50034053240271318757754577619, 6.74339628268830533111895161162, 6.86211201622641231065087748442, 7.52071530922316272284849211141, 7.74027022823604518844485983081, 7.921293429880098905389763300372, 7.958607983584704688710525782500, 8.389949470308854772880028733866
