| L(s) = 1 | + 2·2-s + 2·4-s + 5·5-s − 2·7-s + 5·8-s + 10·10-s + 2·11-s + 9·13-s − 4·14-s + 5·16-s − 8·17-s − 5·19-s + 10·20-s + 4·22-s + 11·23-s + 10·25-s + 18·26-s − 4·28-s − 5·29-s + 3·31-s − 2·32-s − 16·34-s − 10·35-s − 7·37-s − 10·38-s + 25·40-s − 8·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 2.23·5-s − 0.755·7-s + 1.76·8-s + 3.16·10-s + 0.603·11-s + 2.49·13-s − 1.06·14-s + 5/4·16-s − 1.94·17-s − 1.14·19-s + 2.23·20-s + 0.852·22-s + 2.29·23-s + 2·25-s + 3.53·26-s − 0.755·28-s − 0.928·29-s + 0.538·31-s − 0.353·32-s − 2.74·34-s − 1.69·35-s − 1.15·37-s − 1.62·38-s + 3.95·40-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.031943019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.031943019\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_4$ | \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - p T + p T^{2} - 5 T^{3} + 11 T^{4} - 5 p T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 34 T^{3} + 225 T^{4} - 34 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 9 T + 23 T^{2} - 15 T^{3} + 16 T^{4} - 15 p T^{5} + 23 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 8 T + 7 T^{2} - 110 T^{3} - 579 T^{4} - 110 p T^{5} + 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 5 T + 21 T^{2} + 145 T^{3} + 956 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 11 T + 28 T^{2} + 245 T^{3} - 2259 T^{4} + 245 p T^{5} + 28 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 5 T - 19 T^{2} - 5 p T^{3} - 4 T^{4} - 5 p^{2} T^{5} - 19 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 3 T - 22 T^{2} + 159 T^{3} + 205 T^{4} + 159 p T^{5} - 22 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 7 T - 18 T^{2} - 145 T^{3} + 371 T^{4} - 145 p T^{5} - 18 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 8 T - 17 T^{2} - 254 T^{3} - 435 T^{4} - 254 p T^{5} - 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 3 T + 77 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 2 T - 43 T^{2} - 50 T^{3} + 2351 T^{4} - 50 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 9 T + 8 T^{2} + 315 T^{3} + 5131 T^{4} + 315 p T^{5} + 8 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 31 T^{2} - 210 T^{3} + 2851 T^{4} - 210 p T^{5} + 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 13 T + 78 T^{2} - 941 T^{3} + 11075 T^{4} - 941 p T^{5} + 78 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 2 T - 3 T^{2} - 410 T^{3} + 1601 T^{4} - 410 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 8 T - 37 T^{2} - 694 T^{3} - 2425 T^{4} - 694 p T^{5} - 37 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 - 9 T + 8 T^{2} + 585 T^{3} - 5849 T^{4} + 585 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 15 T + 21 T^{2} + 145 T^{3} + 2916 T^{4} + 145 p T^{5} + 21 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 9 T - 52 T^{2} - 675 T^{3} + 121 T^{4} - 675 p T^{5} - 52 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 - 20 T + 151 T^{2} - 1600 T^{3} + 21441 T^{4} - 1600 p T^{5} + 151 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 8 T - 63 T^{2} + 20 T^{3} + 9821 T^{4} + 20 p T^{5} - 63 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−9.251094868300608055779950772959, −8.646476486198490560046529505835, −8.319162286243845419043953168407, −8.282364203319938659883156442403, −8.222942394282048637108043109441, −7.21597218966254580881205878998, −7.12885805024586997477797125196, −6.87077533599086226292170746110, −6.68447540346646701823731038795, −6.32326455311551958490371033256, −6.31247644315263619326520085801, −5.92367509451255423554146422096, −5.80886779301057295495269907811, −5.06338717477443100141975868325, −5.04019110117545233045493522709, −4.86224161834378006994345301739, −4.45854717219692928751856551430, −4.12961664661698091129487997365, −3.52432800886435025671048143092, −3.47644320713367639394951751067, −3.18342651284219635373576048364, −2.38989452429082209185445301018, −1.92366331863080516103889970660, −1.84206524063358666703503926607, −1.32859984070497860860220619301,
1.32859984070497860860220619301, 1.84206524063358666703503926607, 1.92366331863080516103889970660, 2.38989452429082209185445301018, 3.18342651284219635373576048364, 3.47644320713367639394951751067, 3.52432800886435025671048143092, 4.12961664661698091129487997365, 4.45854717219692928751856551430, 4.86224161834378006994345301739, 5.04019110117545233045493522709, 5.06338717477443100141975868325, 5.80886779301057295495269907811, 5.92367509451255423554146422096, 6.31247644315263619326520085801, 6.32326455311551958490371033256, 6.68447540346646701823731038795, 6.87077533599086226292170746110, 7.12885805024586997477797125196, 7.21597218966254580881205878998, 8.222942394282048637108043109441, 8.282364203319938659883156442403, 8.319162286243845419043953168407, 8.646476486198490560046529505835, 9.251094868300608055779950772959
