| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.460462104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.460462104\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 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
−7.65913339828836832565532654024, −7.65429634246921221814901801122, −7.42671116477326197263980006442, −7.20226220740905371285949491999, −6.89858435197585086374222599199, −6.31983640096746890680335091522, −6.20514548278450681386118297719, −6.00678235773738969096745108412, −5.84083781942698524420119909191, −5.77296498686735187792720309104, −5.21591744697219998737670193448, −5.12877884704281348068737638064, −4.79871513576928201258031087100, −4.66578558696794469330022086297, −3.93604656073639836598130336332, −3.88671544927220471547075621835, −3.80036155467176529376893035210, −3.51280139105421303898109979248, −3.47148343844777682612497715294, −2.52688002110971297911429488291, −2.03853526150052686716080141763, −1.86449883122570750352496448683, −1.77476745804642224710503056426, −0.75983506415339821548267893998, −0.58943952282039319654075048682,
0.58943952282039319654075048682, 0.75983506415339821548267893998, 1.77476745804642224710503056426, 1.86449883122570750352496448683, 2.03853526150052686716080141763, 2.52688002110971297911429488291, 3.47148343844777682612497715294, 3.51280139105421303898109979248, 3.80036155467176529376893035210, 3.88671544927220471547075621835, 3.93604656073639836598130336332, 4.66578558696794469330022086297, 4.79871513576928201258031087100, 5.12877884704281348068737638064, 5.21591744697219998737670193448, 5.77296498686735187792720309104, 5.84083781942698524420119909191, 6.00678235773738969096745108412, 6.20514548278450681386118297719, 6.31983640096746890680335091522, 6.89858435197585086374222599199, 7.20226220740905371285949491999, 7.42671116477326197263980006442, 7.65429634246921221814901801122, 7.65913339828836832565532654024
