| L(s) = 1 | + 2·3-s + 4·5-s + 9-s − 4·11-s + 8·15-s + 12·17-s − 8·19-s − 4·23-s + 12·25-s − 2·27-s − 8·33-s − 8·37-s − 24·41-s + 4·45-s − 8·47-s + 24·51-s + 4·53-s − 16·55-s − 16·57-s + 8·61-s − 8·69-s + 8·71-s + 24·73-s + 24·75-s + 16·79-s − 4·81-s − 16·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 1/3·9-s − 1.20·11-s + 2.06·15-s + 2.91·17-s − 1.83·19-s − 0.834·23-s + 12/5·25-s − 0.384·27-s − 1.39·33-s − 1.31·37-s − 3.74·41-s + 0.596·45-s − 1.16·47-s + 3.36·51-s + 0.549·53-s − 2.15·55-s − 2.11·57-s + 1.02·61-s − 0.963·69-s + 0.949·71-s + 2.80·73-s + 2.77·75-s + 1.80·79-s − 4/9·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.623148297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.623148297\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 4 T + 4 T^{2} - 8 T^{3} + 39 T^{4} - 8 p T^{5} + 4 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 16 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T + 76 T^{2} - 24 p T^{3} + 111 p T^{4} - 24 p^{2} T^{5} + 76 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 8 T + 18 T^{2} + 64 T^{3} + 539 T^{4} + 64 p T^{5} + 18 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 4 T - 26 T^{2} - 16 T^{3} + 867 T^{4} - 16 p T^{5} - 26 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 54 T^{2} + 1955 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 8 T + 6 T^{2} - 128 T^{3} - 373 T^{4} - 128 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 100 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8 T + 26 T^{2} - 448 T^{3} - 3773 T^{4} - 448 p T^{5} + 26 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 46 T^{2} - 1365 T^{4} - 46 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 24 T^{2} + 272 T^{3} + 119 T^{4} + 272 p T^{5} - 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^3$ | \( 1 - 6 T^{2} - 4453 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 24 T + 304 T^{2} - 3024 T^{3} + 26607 T^{4} - 3024 p T^{5} + 304 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 16 T + 66 T^{2} - 512 T^{3} + 9635 T^{4} - 512 p T^{5} + 66 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 20 T + 140 T^{2} - 1640 T^{3} + 24079 T^{4} - 1640 p T^{5} + 140 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 8 T + 192 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.42104170246016069615934109289, −6.13926056678923385846675956234, −6.03716766738092625966123426370, −5.79611949242956856891820242829, −5.34371931563698391740734838255, −5.22166565558260161416138467365, −5.18631134901648262582703086394, −5.09952121097565407255483421707, −4.92681701256126270923876143523, −4.63588650196639138261257921762, −4.05471275423980871607708193926, −3.85056203656841762425579502241, −3.61541654655127987695948185593, −3.56691575099585083452908055408, −3.48152635749259623674752739242, −2.86574559246936318204052415230, −2.82178667572197851195289853302, −2.55332808149320120188506239331, −2.38607773969250771241052319631, −1.94893708431035641060370700923, −1.89781058372067096481678535076, −1.55942041022311621935613883421, −1.27913780507361928967728227294, −0.907955569593034552632560229170, −0.15613905863604332750944544085,
0.15613905863604332750944544085, 0.907955569593034552632560229170, 1.27913780507361928967728227294, 1.55942041022311621935613883421, 1.89781058372067096481678535076, 1.94893708431035641060370700923, 2.38607773969250771241052319631, 2.55332808149320120188506239331, 2.82178667572197851195289853302, 2.86574559246936318204052415230, 3.48152635749259623674752739242, 3.56691575099585083452908055408, 3.61541654655127987695948185593, 3.85056203656841762425579502241, 4.05471275423980871607708193926, 4.63588650196639138261257921762, 4.92681701256126270923876143523, 5.09952121097565407255483421707, 5.18631134901648262582703086394, 5.22166565558260161416138467365, 5.34371931563698391740734838255, 5.79611949242956856891820242829, 6.03716766738092625966123426370, 6.13926056678923385846675956234, 6.42104170246016069615934109289
