| L(s) = 1 | − 8·9-s − 10·11-s − 18·23-s − 10·25-s + 6·29-s + 22·37-s − 18·43-s + 18·53-s − 42·67-s − 70·71-s − 6·79-s + 25·81-s + 80·99-s − 34·107-s − 34·109-s − 34·113-s − 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 66·169-s + ⋯ |
| L(s) = 1 | − 8/3·9-s − 3.01·11-s − 3.75·23-s − 2·25-s + 1.11·29-s + 3.61·37-s − 2.74·43-s + 2.47·53-s − 5.13·67-s − 8.30·71-s − 0.675·79-s + 25/9·81-s + 8.04·99-s − 3.28·107-s − 3.25·109-s − 3.19·113-s − 0.272·121-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 − 5.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{18}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 8 T^{2} + 13 p T^{4} + 136 T^{6} + 13 p^{3} T^{8} + 8 p^{4} T^{10} + p^{6} T^{12} \) |
| 5 | \( 1 + 2 p T^{2} + 43 T^{4} + 172 T^{6} + 43 p^{2} T^{8} + 2 p^{5} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 5 T + 39 T^{2} + 111 T^{3} + 39 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 + 66 T^{2} + 1931 T^{4} + 32284 T^{6} + 1931 p^{2} T^{8} + 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 + 4 p T^{2} + 2063 T^{4} + 40656 T^{6} + 2063 p^{2} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} \) |
| 19 | \( 1 + 62 T^{2} + 2075 T^{4} + 46452 T^{6} + 2075 p^{2} T^{8} + 62 p^{4} T^{10} + p^{6} T^{12} \) |
| 23 | \( ( 1 + 9 T + 89 T^{2} + 413 T^{3} + 89 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 29 | \( ( 1 - 3 T + 41 T^{2} - 77 T^{3} + 41 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( 1 + 102 T^{2} + 5763 T^{4} + 220452 T^{6} + 5763 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 37 | \( ( 1 - 11 T + 121 T^{2} - 785 T^{3} + 121 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( 1 + 68 T^{2} + 4783 T^{4} + 172400 T^{6} + 4783 p^{2} T^{8} + 68 p^{4} T^{10} + p^{6} T^{12} \) |
| 43 | \( ( 1 + 9 T + 107 T^{2} + 703 T^{3} + 107 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 47 | \( 1 + 102 T^{2} + 8723 T^{4} + 447940 T^{6} + 8723 p^{2} T^{8} + 102 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( ( 1 - 9 T + 137 T^{2} - 785 T^{3} + 137 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 59 | \( 1 + 104 T^{2} + 7319 T^{4} + 321864 T^{6} + 7319 p^{2} T^{8} + 104 p^{4} T^{10} + p^{6} T^{12} \) |
| 61 | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{3} \) |
| 67 | \( ( 1 + 21 T + 327 T^{2} + 3003 T^{3} + 327 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 71 | \( ( 1 + 35 T + 605 T^{2} + 6349 T^{3} + 605 p T^{4} + 35 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 160 T^{2} + 16755 T^{4} - 1307968 T^{6} + 16755 p^{2} T^{8} - 160 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 3 T + 191 T^{2} + 377 T^{3} + 191 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 + 94 T^{2} + 12907 T^{4} + 1359316 T^{6} + 12907 p^{2} T^{8} + 94 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( 1 + 272 T^{2} + 42899 T^{4} + 4696608 T^{6} + 42899 p^{2} T^{8} + 272 p^{4} T^{10} + p^{6} T^{12} \) |
| 97 | \( 1 + 156 T^{2} + 6239 T^{4} - 161984 T^{6} + 6239 p^{2} T^{8} + 156 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.46199807298304791601762084099, −4.35480354094261674119232009897, −4.19575764242801400763086668971, −4.15756737512715379088059639035, −4.14965278981672016231926221204, −4.12556168911744560882806754717, −3.89444245755597551889307273843, −3.44347692321275022614155750309, −3.25998483638556877761303502242, −3.22002871122289504292140265951, −3.19433393468783765025537770518, −3.10533246880563468206429867756, −2.83140682699582597343210241690, −2.66050270764056668234419269691, −2.62258815217932423572633065480, −2.45130803226677730917002712187, −2.42991165026054946448269631962, −2.24705442090601228339710598083, −2.17189351724531679801181430437, −1.74996462626307074655047381390, −1.67743509226445391288100792776, −1.43808428075982025499304677020, −1.27315351151565958352272989323, −1.10751954379526704092624619307, −0.958005548964247243031098471981, 0, 0, 0, 0, 0, 0,
0.958005548964247243031098471981, 1.10751954379526704092624619307, 1.27315351151565958352272989323, 1.43808428075982025499304677020, 1.67743509226445391288100792776, 1.74996462626307074655047381390, 2.17189351724531679801181430437, 2.24705442090601228339710598083, 2.42991165026054946448269631962, 2.45130803226677730917002712187, 2.62258815217932423572633065480, 2.66050270764056668234419269691, 2.83140682699582597343210241690, 3.10533246880563468206429867756, 3.19433393468783765025537770518, 3.22002871122289504292140265951, 3.25998483638556877761303502242, 3.44347692321275022614155750309, 3.89444245755597551889307273843, 4.12556168911744560882806754717, 4.14965278981672016231926221204, 4.15756737512715379088059639035, 4.19575764242801400763086668971, 4.35480354094261674119232009897, 4.46199807298304791601762084099
Plot not available for L-functions of degree greater than 10.
