| L(s) = 1 | + 4·2-s − 4·3-s + 10·4-s − 16·6-s + 20·8-s + 8·9-s − 12·11-s − 40·12-s + 35·16-s − 24·17-s + 32·18-s − 48·22-s − 80·24-s − 2·25-s − 12·27-s − 16·29-s − 16·31-s + 56·32-s + 48·33-s − 96·34-s + 80·36-s − 16·41-s − 120·44-s − 140·48-s − 2·49-s − 8·50-s + 96·51-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s − 6.53·6-s + 7.07·8-s + 8/3·9-s − 3.61·11-s − 11.5·12-s + 35/4·16-s − 5.82·17-s + 7.54·18-s − 10.2·22-s − 16.3·24-s − 2/5·25-s − 2.30·27-s − 2.97·29-s − 2.87·31-s + 9.89·32-s + 8.35·33-s − 16.4·34-s + 40/3·36-s − 2.49·41-s − 18.0·44-s − 20.2·48-s − 2/7·49-s − 1.13·50-s + 13.4·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| good | 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 66 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 12 T + 4 p T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_4\times C_2$ | \( 1 - 68 T^{2} + 2086 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 76 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 9046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 12130 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 4246 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 706 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 18130 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 216 T^{2} + 22034 T^{4} - 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.79001677352354156557010658112, −6.40318702749952050787373094608, −6.24442177772809096784932775531, −5.96918519113663887818558536533, −5.87395070136540394574941943930, −5.52808118834120429760738682833, −5.51208994723556783493351227546, −5.45694831351818792503386113434, −5.22361876697336580069539705825, −4.88709933198078772139326126983, −4.67091424980643561795791708388, −4.64248462756593832058402128145, −4.58366279642860771931414287961, −4.15813326060230657433909482608, −3.92326752619759412335887674581, −3.68183406984193744591738015550, −3.63439173177617295695501921936, −3.18277547183074997119217515298, −2.76900583716436612530939325115, −2.62568891829270375265740926058, −2.27693943019107075826401901297, −2.21886442683209112316902075158, −1.94986136849685488340663482922, −1.73901622704912851425749804825, −1.41705854565158552541330519286, 0, 0, 0, 0,
1.41705854565158552541330519286, 1.73901622704912851425749804825, 1.94986136849685488340663482922, 2.21886442683209112316902075158, 2.27693943019107075826401901297, 2.62568891829270375265740926058, 2.76900583716436612530939325115, 3.18277547183074997119217515298, 3.63439173177617295695501921936, 3.68183406984193744591738015550, 3.92326752619759412335887674581, 4.15813326060230657433909482608, 4.58366279642860771931414287961, 4.64248462756593832058402128145, 4.67091424980643561795791708388, 4.88709933198078772139326126983, 5.22361876697336580069539705825, 5.45694831351818792503386113434, 5.51208994723556783493351227546, 5.52808118834120429760738682833, 5.87395070136540394574941943930, 5.96918519113663887818558536533, 6.24442177772809096784932775531, 6.40318702749952050787373094608, 6.79001677352354156557010658112
