| L(s) = 1 | − 6·5-s + 9-s + 8·13-s + 19·25-s − 24·37-s − 6·45-s − 11·49-s − 18·53-s + 16·61-s − 48·65-s + 48·73-s + 12·101-s + 12·109-s − 24·113-s + 8·117-s − 5·121-s − 66·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 1/3·9-s + 2.21·13-s + 19/5·25-s − 3.94·37-s − 0.894·45-s − 1.57·49-s − 2.47·53-s + 2.04·61-s − 5.95·65-s + 5.61·73-s + 1.19·101-s + 1.14·109-s − 2.25·113-s + 0.739·117-s − 0.454·121-s − 5.90·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.628119018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628119018\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 5 T^{2} - 96 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 59 T^{2} + p^{2} T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 9 T + 80 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 37 T^{2} - 2112 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^3$ | \( 1 + 133 T^{2} + 11448 T^{4} + 133 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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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
−6.78282165062206622191406921291, −6.66913816317918889745690932381, −6.60708691869720987702872149222, −6.35707414233252814269963233906, −5.97429061832092713126736282267, −5.93260039119670303374738713895, −5.36168345650357830301906189467, −5.15450605305712353113277637041, −4.98703405998182060370574010417, −4.95593138404392457615526334723, −4.72117772292414441265933850874, −4.07553776116679923193109711420, −4.01796627633448204520315930169, −3.82038357252762408895905176482, −3.79771425443610028903840505899, −3.37855125958545377966365934512, −3.29651242218916305661339182808, −3.16064146809860865262126742855, −2.64075685110618720270407757723, −2.22740284118668090368785118892, −1.76897921565750791015094913909, −1.59531325911626369519157234188, −1.23425780062628241436241687421, −0.52339953347459168948802790066, −0.47500003588282777415741738689,
0.47500003588282777415741738689, 0.52339953347459168948802790066, 1.23425780062628241436241687421, 1.59531325911626369519157234188, 1.76897921565750791015094913909, 2.22740284118668090368785118892, 2.64075685110618720270407757723, 3.16064146809860865262126742855, 3.29651242218916305661339182808, 3.37855125958545377966365934512, 3.79771425443610028903840505899, 3.82038357252762408895905176482, 4.01796627633448204520315930169, 4.07553776116679923193109711420, 4.72117772292414441265933850874, 4.95593138404392457615526334723, 4.98703405998182060370574010417, 5.15450605305712353113277637041, 5.36168345650357830301906189467, 5.93260039119670303374738713895, 5.97429061832092713126736282267, 6.35707414233252814269963233906, 6.60708691869720987702872149222, 6.66913816317918889745690932381, 6.78282165062206622191406921291
