| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 4·6-s + 4·8-s + 9-s + 2·11-s − 4·12-s − 16·13-s − 12·16-s + 10·17-s − 2·18-s − 2·19-s − 4·22-s − 6·23-s − 8·24-s + 32·26-s + 2·27-s + 4·29-s − 6·31-s + 16·32-s − 4·33-s − 20·34-s + 2·36-s + 4·37-s + 4·38-s + 32·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s + 1.41·8-s + 1/3·9-s + 0.603·11-s − 1.15·12-s − 4.43·13-s − 3·16-s + 2.42·17-s − 0.471·18-s − 0.458·19-s − 0.852·22-s − 1.25·23-s − 1.63·24-s + 6.27·26-s + 0.384·27-s + 0.742·29-s − 1.07·31-s + 2.82·32-s − 0.696·33-s − 3.42·34-s + 1/3·36-s + 0.657·37-s + 0.648·38-s + 5.12·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09321725046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09321725046\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T + p T^{2} )^{2}( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 4 T^{3} + 235 T^{4} + 4 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 3 p T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10 T + 44 T^{2} - 220 T^{3} + 1147 T^{4} - 220 p T^{5} + 44 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 23 T^{2} - 22 T^{3} + 292 T^{4} - 22 p T^{5} - 23 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 6 T - 16 T^{2} + 36 T^{3} + 1347 T^{4} + 36 p T^{5} - 16 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 18 T^{3} + 1404 T^{4} - 18 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 35 T^{2} + 92 T^{3} + 640 T^{4} + 92 p T^{5} - 35 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 2 T + 80 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 4 T + 63 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4 T + 14 T^{2} + 416 T^{3} - 3653 T^{4} + 416 p T^{5} + 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 10 T - 16 T^{2} - 20 T^{3} + 4075 T^{4} - 20 p T^{5} - 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 49 T^{2} - 468 T^{3} - 5112 T^{4} - 468 p T^{5} + 49 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T - 23 T^{2} + 472 T^{3} - 2432 T^{4} + 472 p T^{5} - 23 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T - 23 T^{2} - 594 T^{3} - 5604 T^{4} - 594 p T^{5} - 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 4 T^{2} - 828 T^{3} - 9525 T^{4} - 828 p T^{5} - 4 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.76051782627422204414350666907, −7.68703974669635499769986208650, −7.45816648713821278374193928382, −7.15963689908187524841796891918, −6.98123974872391439139782675325, −6.97672055338985661064281058813, −6.35127040971011091124678852270, −6.12767522862625567105627362265, −6.12766302616740964396092817931, −5.35336596357687700716879020563, −5.29895431281522139939526090243, −5.22959840953971991922015199959, −5.09085576384468473926374409261, −4.47691523437194824026953194871, −4.26160705236712672132013570803, −4.16772047827140247390038108307, −4.04885955873784853450992779165, −3.02780322225673365980808133131, −2.93910779492461082331952835779, −2.69733941925457171605586988936, −2.16602590777935336527781732797, −1.84943518277618743964910931655, −1.43164456206703976689329821328, −1.00669707170838769025332465468, −0.15905344153980337818924475531,
0.15905344153980337818924475531, 1.00669707170838769025332465468, 1.43164456206703976689329821328, 1.84943518277618743964910931655, 2.16602590777935336527781732797, 2.69733941925457171605586988936, 2.93910779492461082331952835779, 3.02780322225673365980808133131, 4.04885955873784853450992779165, 4.16772047827140247390038108307, 4.26160705236712672132013570803, 4.47691523437194824026953194871, 5.09085576384468473926374409261, 5.22959840953971991922015199959, 5.29895431281522139939526090243, 5.35336596357687700716879020563, 6.12766302616740964396092817931, 6.12767522862625567105627362265, 6.35127040971011091124678852270, 6.97672055338985661064281058813, 6.98123974872391439139782675325, 7.15963689908187524841796891918, 7.45816648713821278374193928382, 7.68703974669635499769986208650, 7.76051782627422204414350666907
