| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 5·9-s + 4·11-s + 8·13-s − 4·15-s + 4·17-s − 4·21-s + 14·23-s + 25-s + 10·27-s − 20·29-s + 4·31-s + 8·33-s + 4·35-s + 16·39-s + 12·41-s + 12·43-s − 10·45-s − 12·47-s + 7·49-s + 8·51-s − 8·55-s + 8·59-s − 2·61-s − 10·63-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 5/3·9-s + 1.20·11-s + 2.21·13-s − 1.03·15-s + 0.970·17-s − 0.872·21-s + 2.91·23-s + 1/5·25-s + 1.92·27-s − 3.71·29-s + 0.718·31-s + 1.39·33-s + 0.676·35-s + 2.56·39-s + 1.87·41-s + 1.82·43-s − 1.49·45-s − 1.75·47-s + 49-s + 1.12·51-s − 1.07·55-s + 1.04·59-s − 0.256·61-s − 1.25·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.367913427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.367913427\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_4\times C_2$ | \( 1 - 4 T + 10 T^{2} + 112 T^{3} - 525 T^{4} + 112 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 6 T^{2} - 325 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 14 T + 103 T^{2} - 658 T^{3} + 3612 T^{4} - 658 p T^{5} + 103 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 75 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 42 T^{2} + 16 T^{3} + 1907 T^{4} + 16 p T^{5} - 42 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 42 T^{2} + 395 T^{4} - 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 6 T - 3 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 46 T^{2} + 48 T^{3} + 627 T^{4} + 48 p T^{5} + 46 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 74 T^{2} + 2667 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 87 T^{2} - 62 T^{3} + 4316 T^{4} - 62 p T^{5} - 87 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T - 9 T^{2} + 534 T^{3} - 4876 T^{4} + 534 p T^{5} - 9 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 102 T^{2} + 112 T^{3} + 7427 T^{4} + 112 p T^{5} - 102 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 18 T + 229 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 22 T + 217 T^{2} - 22 p T^{3} + 228 p T^{4} - 22 p^{2} T^{5} + 217 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
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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
−8.752062120743038113528435166263, −8.673342836698844699501629770381, −7.990939318172307776076728691178, −7.77056097420570372136053467131, −7.71611618647286109197631231783, −7.33317776428191700037641836602, −7.20226861002066741457509607574, −6.92504814515629327873159953462, −6.69957839915771996295503766977, −6.24846153389133326059142244730, −5.93320887717007170126663228607, −5.88126113513778346760081005637, −5.50931223384563075237847348737, −5.06629752891740358246001047760, −4.56465708627764315012572501010, −4.38171337426815920232364466510, −3.96709219272168696580941660623, −3.79191535636515774799593132944, −3.52318138277711229973587567654, −3.34266005373798626897546140278, −2.82777380518921199031991292441, −2.61710245773607030521634850411, −1.75438787182776435296767405066, −1.24851763638738315928648747943, −1.08428940480459934603475646725,
1.08428940480459934603475646725, 1.24851763638738315928648747943, 1.75438787182776435296767405066, 2.61710245773607030521634850411, 2.82777380518921199031991292441, 3.34266005373798626897546140278, 3.52318138277711229973587567654, 3.79191535636515774799593132944, 3.96709219272168696580941660623, 4.38171337426815920232364466510, 4.56465708627764315012572501010, 5.06629752891740358246001047760, 5.50931223384563075237847348737, 5.88126113513778346760081005637, 5.93320887717007170126663228607, 6.24846153389133326059142244730, 6.69957839915771996295503766977, 6.92504814515629327873159953462, 7.20226861002066741457509607574, 7.33317776428191700037641836602, 7.71611618647286109197631231783, 7.77056097420570372136053467131, 7.990939318172307776076728691178, 8.673342836698844699501629770381, 8.752062120743038113528435166263
