| L(s) = 1 | − 2·3-s − 2·4-s + 4·5-s + 6·7-s + 2·9-s + 4·12-s − 8·15-s + 3·16-s + 12·17-s − 8·20-s − 12·21-s + 10·25-s − 6·27-s − 12·28-s + 24·35-s − 4·36-s + 12·37-s + 8·41-s + 16·43-s + 8·45-s − 8·47-s − 6·48-s + 18·49-s − 24·51-s + 16·60-s + 12·63-s − 4·64-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 1.78·5-s + 2.26·7-s + 2/3·9-s + 1.15·12-s − 2.06·15-s + 3/4·16-s + 2.91·17-s − 1.78·20-s − 2.61·21-s + 2·25-s − 1.15·27-s − 2.26·28-s + 4.05·35-s − 2/3·36-s + 1.97·37-s + 1.24·41-s + 2.43·43-s + 1.19·45-s − 1.16·47-s − 0.866·48-s + 18/7·49-s − 3.36·51-s + 2.06·60-s + 1.51·63-s − 1/2·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.851224971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.851224971\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| good | 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 | $D_{4}$ | \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 15406 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 2 T - 78 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−9.274618619610422215752940428211, −8.938106051143470080906408762661, −8.494393243429295623907305615873, −8.111410762312666473310373948238, −7.919321663811144579775381784087, −7.88102971389047702579319886662, −7.52206610716490829259471236966, −7.21880171673753600146579085201, −6.98616253069062895879214867105, −6.22383389065855848885223247386, −6.21026672455125969005904546329, −5.78020523149887817724904724686, −5.56250779248751694961822887916, −5.49291523962495038756578242293, −5.39378971821176985731350793601, −4.85894466694509251136112359601, −4.37719302623411572614881697721, −4.25110150697653622959183206661, −4.24058733389538653244322784739, −3.33016707163840279440937490097, −2.72901789556295232313364175973, −2.70748439083832360578983353683, −1.70144517042571448310252289478, −1.29525487535678039644420450590, −1.21527653637586685339144657534,
1.21527653637586685339144657534, 1.29525487535678039644420450590, 1.70144517042571448310252289478, 2.70748439083832360578983353683, 2.72901789556295232313364175973, 3.33016707163840279440937490097, 4.24058733389538653244322784739, 4.25110150697653622959183206661, 4.37719302623411572614881697721, 4.85894466694509251136112359601, 5.39378971821176985731350793601, 5.49291523962495038756578242293, 5.56250779248751694961822887916, 5.78020523149887817724904724686, 6.21026672455125969005904546329, 6.22383389065855848885223247386, 6.98616253069062895879214867105, 7.21880171673753600146579085201, 7.52206610716490829259471236966, 7.88102971389047702579319886662, 7.919321663811144579775381784087, 8.111410762312666473310373948238, 8.494393243429295623907305615873, 8.938106051143470080906408762661, 9.274618619610422215752940428211
