L(s) = 1 | + 2·3-s + 4·5-s + 2·9-s + 8·15-s − 20·19-s − 16·23-s + 6·27-s − 16·43-s + 8·45-s + 8·47-s − 2·49-s − 24·53-s − 40·57-s + 8·67-s − 32·69-s − 8·71-s + 16·73-s + 11·81-s − 80·95-s + 32·97-s − 52·101-s − 64·115-s + 4·121-s − 20·125-s + 127-s − 32·129-s + 131-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 1.78·5-s + 2/3·9-s + 2.06·15-s − 4.58·19-s − 3.33·23-s + 1.15·27-s − 2.43·43-s + 1.19·45-s + 1.16·47-s − 2/7·49-s − 3.29·53-s − 5.29·57-s + 0.977·67-s − 3.85·69-s − 0.949·71-s + 1.87·73-s + 11/9·81-s − 8.20·95-s + 3.24·97-s − 5.17·101-s − 5.96·115-s + 4/11·121-s − 1.78·125-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5792475964\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5792475964\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 5 | $D_{4}$ | \( ( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 4918 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 12 T + 122 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 19486 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 5646 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_4$ | \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 11206 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 12958 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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
−7.70321722261000367576773512329, −7.34887582644945276889519399589, −7.10051262179613694444387755747, −6.59919948266156650455513040904, −6.48655236715676557352128848136, −6.34919369670610340982207952401, −6.26274766130226540521493806937, −6.12903207696050279927386826633, −5.70671892910726035786767975308, −5.54088473557920677750225394549, −5.06075724820544981892478111924, −4.92841506400247121958072650782, −4.42199383630428046038802185386, −4.41155465914585970432949592283, −4.08642522780298187312639876311, −3.77760617853955280839338698777, −3.59011774973407130960702526833, −3.25660599623607248556918686877, −2.55494884535674399791079643881, −2.50391266319165220192892077472, −2.28012175826183377934055969070, −1.85058141211833865736529027809, −1.76525964542637424627347857019, −1.54972273034918846143532238428, −0.15941993248579120173184161795,
0.15941993248579120173184161795, 1.54972273034918846143532238428, 1.76525964542637424627347857019, 1.85058141211833865736529027809, 2.28012175826183377934055969070, 2.50391266319165220192892077472, 2.55494884535674399791079643881, 3.25660599623607248556918686877, 3.59011774973407130960702526833, 3.77760617853955280839338698777, 4.08642522780298187312639876311, 4.41155465914585970432949592283, 4.42199383630428046038802185386, 4.92841506400247121958072650782, 5.06075724820544981892478111924, 5.54088473557920677750225394549, 5.70671892910726035786767975308, 6.12903207696050279927386826633, 6.26274766130226540521493806937, 6.34919369670610340982207952401, 6.48655236715676557352128848136, 6.59919948266156650455513040904, 7.10051262179613694444387755747, 7.34887582644945276889519399589, 7.70321722261000367576773512329