| L(s) = 1 | + 5-s − 11-s − 10·13-s + 8·17-s − 5·19-s + 8·23-s − 4·25-s − 6·29-s − 2·31-s − 3·37-s + 12·41-s − 14·43-s + 12·47-s + 11·53-s − 55-s − 5·59-s + 20·61-s − 10·65-s − 7·67-s − 8·71-s − 73-s − 8·79-s + 14·83-s + 8·85-s + 6·89-s − 5·95-s − 50·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s − 2.77·13-s + 1.94·17-s − 1.14·19-s + 1.66·23-s − 4/5·25-s − 1.11·29-s − 0.359·31-s − 0.493·37-s + 1.87·41-s − 2.13·43-s + 1.75·47-s + 1.51·53-s − 0.134·55-s − 0.650·59-s + 2.56·61-s − 1.24·65-s − 0.855·67-s − 0.949·71-s − 0.117·73-s − 0.900·79-s + 1.53·83-s + 0.867·85-s + 0.635·89-s − 0.512·95-s − 5.07·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2923022367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2923022367\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 + T - 7 T^{2} - 14 T^{3} - 68 T^{4} - 14 p T^{5} - 7 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 5 T - 5 T^{2} - 40 T^{3} + 64 T^{4} - 40 p T^{5} - 5 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 3 T - 53 T^{2} - 36 T^{3} + 2142 T^{4} - 36 p T^{5} - 53 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 7 T + 84 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11 T - T^{2} - 176 T^{3} + 5662 T^{4} - 176 p T^{5} - p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 5 T - 85 T^{2} - 40 T^{3} + 7144 T^{4} - 40 p T^{5} - 85 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 7 T - 83 T^{2} - 14 T^{3} + 9652 T^{4} - 14 p T^{5} - 83 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + T - 131 T^{2} - 14 T^{3} + 12022 T^{4} - 14 p T^{5} - 131 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 53 T^{2} - 328 T^{3} + 2392 T^{4} - 328 p T^{5} - 53 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 7 T + 164 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 94 T^{2} + 288 T^{3} + 5775 T^{4} + 288 p T^{5} - 94 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 25 T + 336 T^{2} + 25 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.91446325531407474128479664930, −5.80247098895980610771990663890, −5.55680545253185996495740135826, −5.30964477141230028533100347988, −5.30945524856938289681403299200, −5.27206336187944315418307473471, −5.00314489569266071102728237433, −4.68321397039755359850745506829, −4.37311254869851507936916089953, −4.23725055952353828678947301846, −4.12581092191666211063382942849, −3.69721416794052783552388272573, −3.66736694168534495385286422734, −3.41644564917240191441888541885, −3.04221927774372943704200864024, −2.65307693125030121054280548218, −2.54628300369273177746442245964, −2.54293324283844981834855025050, −2.40603133068610929621282892548, −1.85786668033788830292703780484, −1.53527858045962493776003662470, −1.46132342779507892807892236127, −1.01992446524245547949977682468, −0.59850780740861385076229546921, −0.090562831323365481402228476816,
0.090562831323365481402228476816, 0.59850780740861385076229546921, 1.01992446524245547949977682468, 1.46132342779507892807892236127, 1.53527858045962493776003662470, 1.85786668033788830292703780484, 2.40603133068610929621282892548, 2.54293324283844981834855025050, 2.54628300369273177746442245964, 2.65307693125030121054280548218, 3.04221927774372943704200864024, 3.41644564917240191441888541885, 3.66736694168534495385286422734, 3.69721416794052783552388272573, 4.12581092191666211063382942849, 4.23725055952353828678947301846, 4.37311254869851507936916089953, 4.68321397039755359850745506829, 5.00314489569266071102728237433, 5.27206336187944315418307473471, 5.30945524856938289681403299200, 5.30964477141230028533100347988, 5.55680545253185996495740135826, 5.80247098895980610771990663890, 5.91446325531407474128479664930
