| L(s) = 1 | − 4-s − 6·11-s + 16-s + 10·19-s + 14·31-s + 6·41-s + 6·44-s − 4·49-s + 18·59-s + 26·61-s − 5·64-s + 18·71-s − 10·76-s + 4·79-s − 12·89-s − 54·101-s + 16·109-s − 5·121-s − 14·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.80·11-s + 1/4·16-s + 2.29·19-s + 2.51·31-s + 0.937·41-s + 0.904·44-s − 4/7·49-s + 2.34·59-s + 3.32·61-s − 5/8·64-s + 2.13·71-s − 1.14·76-s + 0.450·79-s − 1.27·89-s − 5.37·101-s + 1.53·109-s − 0.454·121-s − 1.25·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.334709740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.334709740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4$ | \( 1 + T^{2} + p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 3 T + 16 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + T^{2} + 264 T^{4} + p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 61 T^{2} + 1500 T^{4} + 61 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 1950 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 85 T^{2} + 3876 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 112 T^{2} + 6366 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 2718 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 9 T + 130 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 13 T + 156 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 64 T^{2} + 8814 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 9 T + 88 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 229 T^{2} + 23100 T^{4} + 229 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 16878 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 + 40 T^{2} - 10482 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.65768787783319802393800337747, −6.33267393318701672883609549420, −5.95199138279401374934314039356, −5.79942762852804608199332922616, −5.53719477193458471242261754997, −5.25342716471809768409104972232, −5.23669007938245678852923053044, −5.19963680589248136776790715480, −5.08624936394520101909453912559, −4.32670216263277471851733607845, −4.28003507268547598856096073066, −4.27391875959344170045112216843, −4.10975677753311819221415088426, −3.52526419063155925093872385580, −3.31463853360213832684398568366, −3.15572607210087292342372784192, −2.93093175839376938299028854248, −2.69054055616450952486931111517, −2.32371804454322799126956888294, −2.26565737197796142338681266266, −1.87647681023972559967246638990, −1.31504907889206821428381877232, −0.938050921581072324313989381296, −0.821169904031362597455117875489, −0.40403531966789307930224693327,
0.40403531966789307930224693327, 0.821169904031362597455117875489, 0.938050921581072324313989381296, 1.31504907889206821428381877232, 1.87647681023972559967246638990, 2.26565737197796142338681266266, 2.32371804454322799126956888294, 2.69054055616450952486931111517, 2.93093175839376938299028854248, 3.15572607210087292342372784192, 3.31463853360213832684398568366, 3.52526419063155925093872385580, 4.10975677753311819221415088426, 4.27391875959344170045112216843, 4.28003507268547598856096073066, 4.32670216263277471851733607845, 5.08624936394520101909453912559, 5.19963680589248136776790715480, 5.23669007938245678852923053044, 5.25342716471809768409104972232, 5.53719477193458471242261754997, 5.79942762852804608199332922616, 5.95199138279401374934314039356, 6.33267393318701672883609549420, 6.65768787783319802393800337747
