| L(s) = 1 | + 4·4-s − 4·7-s + 7·16-s − 16·28-s − 8·37-s − 8·43-s + 24·47-s − 2·49-s − 24·59-s + 8·64-s + 16·67-s − 16·79-s − 24·83-s − 24·89-s + 32·109-s − 28·112-s + 16·121-s + 127-s + 131-s + 137-s + 139-s − 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.51·7-s + 7/4·16-s − 3.02·28-s − 1.31·37-s − 1.21·43-s + 3.50·47-s − 2/7·49-s − 3.12·59-s + 64-s + 1.95·67-s − 1.80·79-s − 2.63·83-s − 2.54·89-s + 3.06·109-s − 2.64·112-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9922990402\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9922990402\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 9 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 114 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 90 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 486 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2934 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 5826 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1206 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 32 T^{2} + 78 p T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25110 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 190 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 15606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.63579756358362180508736033291, −6.53513597041654909270423705602, −6.25842290666714857585715401016, −6.15700566104011414588971634033, −6.09232463331881387388497538460, −5.62233514719012109127896496325, −5.55724036889274050239037891694, −5.27288602014921514800326649622, −5.22155650966124867975715477809, −4.61617643212068070660401879610, −4.32850318315833994956933670068, −4.27069684410793498293823010094, −4.23574759849997614264635815688, −3.40572981782655680787615359705, −3.35805011163688987684933260444, −3.34008550120200504771844289341, −3.16782251333163720374521242951, −2.64007372771614772705347750599, −2.46735124269486973015056308051, −2.35121658388123257591330787944, −1.82296152483570410160246725543, −1.73492033069647161129645230219, −1.29133477430316906681036468165, −0.902393286225419105263988306086, −0.17114046287340713083768345047,
0.17114046287340713083768345047, 0.902393286225419105263988306086, 1.29133477430316906681036468165, 1.73492033069647161129645230219, 1.82296152483570410160246725543, 2.35121658388123257591330787944, 2.46735124269486973015056308051, 2.64007372771614772705347750599, 3.16782251333163720374521242951, 3.34008550120200504771844289341, 3.35805011163688987684933260444, 3.40572981782655680787615359705, 4.23574759849997614264635815688, 4.27069684410793498293823010094, 4.32850318315833994956933670068, 4.61617643212068070660401879610, 5.22155650966124867975715477809, 5.27288602014921514800326649622, 5.55724036889274050239037891694, 5.62233514719012109127896496325, 6.09232463331881387388497538460, 6.15700566104011414588971634033, 6.25842290666714857585715401016, 6.53513597041654909270423705602, 6.63579756358362180508736033291
