| L(s) = 1 | − 2·2-s − 4·3-s + 4-s + 2·5-s + 8·6-s + 2·8-s + 6·9-s − 4·10-s − 4·11-s − 4·12-s − 8·15-s − 4·16-s − 8·17-s − 12·18-s − 20·19-s + 2·20-s + 8·22-s + 2·23-s − 8·24-s + 5·25-s + 4·27-s + 4·29-s + 16·30-s + 12·31-s + 2·32-s + 16·33-s + 16·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 1/2·4-s + 0.894·5-s + 3.26·6-s + 0.707·8-s + 2·9-s − 1.26·10-s − 1.20·11-s − 1.15·12-s − 2.06·15-s − 16-s − 1.94·17-s − 2.82·18-s − 4.58·19-s + 0.447·20-s + 1.70·22-s + 0.417·23-s − 1.63·24-s + 25-s + 0.769·27-s + 0.742·29-s + 2.92·30-s + 2.15·31-s + 0.353·32-s + 2.78·33-s + 2.74·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09654753521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09654753521\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 p T^{3} - 4 p T^{4} + 2 p^{2} T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T - 22 T^{2} + 80 T^{3} + 139 T^{4} + 80 p T^{5} - 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 2 T^{2} - 2205 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 18 T + 127 T^{2} - 1350 T^{3} + 15324 T^{4} - 1350 p T^{5} + 127 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 16 T + 82 T^{2} - 640 T^{3} + 8635 T^{4} - 640 p T^{5} + 82 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 107 T^{2} - 90 T^{3} + 11364 T^{4} - 90 p T^{5} - 107 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T - 158 T^{2} - 80 T^{3} + 19315 T^{4} - 80 p T^{5} - 158 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.97734700912612661737372316926, −6.89932105280854433608667163826, −6.54382890908331006971200496543, −6.53074051385720163525805504821, −6.49971606103954165546420161574, −6.34010757382252975576665366156, −6.09672986022286789740818430773, −5.65250283534070668542712635750, −5.56441367847475566597019357691, −5.04127553823616046254469806565, −4.92818472050756686398556848789, −4.83121448432559159802199722762, −4.53555764127408774226362916045, −4.33219123286941075128343313079, −4.32750700503756852863999086500, −3.55489903639101063916153399937, −3.46079829798118555827923288076, −2.74643705175689561112037603501, −2.35397769189536873890165717798, −2.29754645330093255553574832563, −2.26168676746822973766810741876, −1.63878602336526902549030941580, −0.983699544510332516838379828107, −0.66603612531601536746840957048, −0.19961564789279620915249526008,
0.19961564789279620915249526008, 0.66603612531601536746840957048, 0.983699544510332516838379828107, 1.63878602336526902549030941580, 2.26168676746822973766810741876, 2.29754645330093255553574832563, 2.35397769189536873890165717798, 2.74643705175689561112037603501, 3.46079829798118555827923288076, 3.55489903639101063916153399937, 4.32750700503756852863999086500, 4.33219123286941075128343313079, 4.53555764127408774226362916045, 4.83121448432559159802199722762, 4.92818472050756686398556848789, 5.04127553823616046254469806565, 5.56441367847475566597019357691, 5.65250283534070668542712635750, 6.09672986022286789740818430773, 6.34010757382252975576665366156, 6.49971606103954165546420161574, 6.53074051385720163525805504821, 6.54382890908331006971200496543, 6.89932105280854433608667163826, 6.97734700912612661737372316926
