| L(s) = 1 | − 12·3-s + 12·7-s + 90·9-s − 144·21-s + 104·23-s − 540·27-s − 444·29-s − 96·41-s − 240·43-s + 800·47-s − 864·49-s − 504·61-s + 1.08e3·63-s + 984·67-s − 1.24e3·69-s + 2.83e3·81-s + 800·83-s + 5.32e3·87-s + 936·89-s − 2.43e3·101-s + 972·103-s + 16·107-s − 2.64e3·109-s − 3.94e3·121-s + 1.15e3·123-s + 127-s + 2.88e3·129-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.647·7-s + 10/3·9-s − 1.49·21-s + 0.942·23-s − 3.84·27-s − 2.84·29-s − 0.365·41-s − 0.851·43-s + 2.48·47-s − 2.51·49-s − 1.05·61-s + 2.15·63-s + 1.79·67-s − 2.17·69-s + 35/9·81-s + 1.05·83-s + 6.56·87-s + 1.11·89-s − 2.39·101-s + 0.929·103-s + 0.0144·107-s − 2.31·109-s − 2.96·121-s + 0.844·123-s + 0.000698·127-s + 1.96·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 - 6 T + 486 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 3944 T^{2} + 7243806 T^{4} + 3944 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 3232 T^{2} + 29790 p^{2} T^{4} + 3232 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 8288 T^{2} + 39256830 T^{4} + 8288 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 25180 T^{2} + 252479478 T^{4} + 25180 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 52 T + 17486 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 222 T + 44170 T^{2} + 222 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 35012 T^{2} + 1848403974 T^{4} - 35012 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 1216 p T^{2} + 3402706734 T^{4} + 1216 p^{7} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 48 T + 137582 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 120 T + 109110 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 400 T + 217550 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 62432 T^{2} - 24642960210 T^{4} + 62432 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 250472 T^{2} + 51541498974 T^{4} + 250472 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 252 T - 11698 T^{2} + 252 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 492 T + 641142 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 882380 T^{2} + 411599064198 T^{4} + 882380 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 1302484 T^{2} + 723264446598 T^{4} + 1302484 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 108412 T^{2} + 427397621574 T^{4} + 108412 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 400 T + 912710 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 468 T + 712294 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 3432196 T^{2} + 4607207340678 T^{4} + 3432196 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.50968837384118983829384898696, −6.01213171250142570994240126914, −5.85963637735718507602271599377, −5.79808644841047588498308346141, −5.70761718271014292122308332205, −5.26373045383563648160443702097, −5.16104808289730746976369815802, −5.10239504864285785430588304874, −4.95361188242791660175649800363, −4.45100101948634864898077839768, −4.37511811087211091056030214325, −4.34367713062788806286944007639, −4.00001439124515104123391341029, −3.57612424745336974696966832088, −3.44650832751555627224644834155, −3.28583515782244894064996926876, −3.19734431853980953721732377511, −2.40820556086627264954403155119, −2.18620620612610934886517177389, −2.18522628599136347220749314373, −2.02058192837196068142968395337, −1.29060543130506661258653986242, −1.18165995153092068380175708597, −1.11486306612601878692589583940, −1.00664107738434799043130105361, 0, 0, 0, 0,
1.00664107738434799043130105361, 1.11486306612601878692589583940, 1.18165995153092068380175708597, 1.29060543130506661258653986242, 2.02058192837196068142968395337, 2.18522628599136347220749314373, 2.18620620612610934886517177389, 2.40820556086627264954403155119, 3.19734431853980953721732377511, 3.28583515782244894064996926876, 3.44650832751555627224644834155, 3.57612424745336974696966832088, 4.00001439124515104123391341029, 4.34367713062788806286944007639, 4.37511811087211091056030214325, 4.45100101948634864898077839768, 4.95361188242791660175649800363, 5.10239504864285785430588304874, 5.16104808289730746976369815802, 5.26373045383563648160443702097, 5.70761718271014292122308332205, 5.79808644841047588498308346141, 5.85963637735718507602271599377, 6.01213171250142570994240126914, 6.50968837384118983829384898696
