| L(s) = 1 | + 9-s + 2·25-s − 6·37-s + 49-s + 2·73-s − 8·97-s + 6·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 9-s + 2·25-s − 6·37-s + 49-s + 2·73-s − 8·97-s + 6·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.059346665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059346665\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$ | \( ( 1 + T )^{8} \) |
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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
−7.04443811950031341510544771084, −6.87501905954004038034005068647, −6.77261580839735768819170297878, −6.71263271333190800662654143049, −6.24508700950942968374562384530, −5.84351289002464920713512138878, −5.66656599874951787414939501177, −5.58575262046103692319798004497, −5.38537914834218028578613808840, −4.91970309487637826133600981641, −4.82638861558338474643732850872, −4.73069243349022429596076097372, −4.52946621521874778386516169117, −4.03345799462827841015483671990, −3.73619799099786836484352427627, −3.71094013236821201846506839146, −3.37908656650643418063308204500, −3.21614537750747988439206086846, −2.84874015659539083664884360037, −2.42190489136619635685284795295, −2.32637257126767035231021023640, −1.71936963225688645783096004183, −1.56163922652239484991610100398, −1.40439597545373163190001410182, −0.68323814532125204780336630358,
0.68323814532125204780336630358, 1.40439597545373163190001410182, 1.56163922652239484991610100398, 1.71936963225688645783096004183, 2.32637257126767035231021023640, 2.42190489136619635685284795295, 2.84874015659539083664884360037, 3.21614537750747988439206086846, 3.37908656650643418063308204500, 3.71094013236821201846506839146, 3.73619799099786836484352427627, 4.03345799462827841015483671990, 4.52946621521874778386516169117, 4.73069243349022429596076097372, 4.82638861558338474643732850872, 4.91970309487637826133600981641, 5.38537914834218028578613808840, 5.58575262046103692319798004497, 5.66656599874951787414939501177, 5.84351289002464920713512138878, 6.24508700950942968374562384530, 6.71263271333190800662654143049, 6.77261580839735768819170297878, 6.87501905954004038034005068647, 7.04443811950031341510544771084
