| L(s) = 1 | − 2·3-s + 4-s + 9-s + 2·11-s − 2·12-s + 4·17-s − 4·19-s − 25-s + 2·27-s − 4·33-s + 36-s − 2·41-s + 2·43-s + 2·44-s − 8·51-s + 8·57-s − 64-s + 4·68-s − 4·73-s + 2·75-s − 4·76-s − 4·81-s − 2·83-s − 4·89-s + 2·97-s + 2·99-s − 100-s + ⋯ |
| L(s) = 1 | − 2·3-s + 4-s + 9-s + 2·11-s − 2·12-s + 4·17-s − 4·19-s − 25-s + 2·27-s − 4·33-s + 36-s − 2·41-s + 2·43-s + 2·44-s − 8·51-s + 8·57-s − 64-s + 4·68-s − 4·73-s + 2·75-s − 4·76-s − 4·81-s − 2·83-s − 4·89-s + 2·97-s + 2·99-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1991188755\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1991188755\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.28699124025639060076281459193, −6.15801425521371923824834134501, −5.79921657306849615897391611616, −5.59492169770929319356041996713, −5.53180956507558771160557584707, −5.50809492341416810523843782690, −5.37087692657741975151146833972, −4.76676792954973029526499125662, −4.70525641638961301599220273962, −4.29931682151293741919426009340, −4.25140011418134892422980319928, −4.19112552664528034720605501589, −3.96514661201432798844170104391, −3.40985955572413424981345275680, −3.39864557703299118438646358323, −3.30277537054758196906221565016, −2.77811340387787122638099571724, −2.55651692164476565764715583662, −2.43861747304447788542711629938, −2.19186759521768043955747887430, −1.49614180284212668885195459786, −1.42576062837841225908658133342, −1.27297274609380430027497668110, −1.23217309827898492178071716728, −0.18575091203530133148403206079,
0.18575091203530133148403206079, 1.23217309827898492178071716728, 1.27297274609380430027497668110, 1.42576062837841225908658133342, 1.49614180284212668885195459786, 2.19186759521768043955747887430, 2.43861747304447788542711629938, 2.55651692164476565764715583662, 2.77811340387787122638099571724, 3.30277537054758196906221565016, 3.39864557703299118438646358323, 3.40985955572413424981345275680, 3.96514661201432798844170104391, 4.19112552664528034720605501589, 4.25140011418134892422980319928, 4.29931682151293741919426009340, 4.70525641638961301599220273962, 4.76676792954973029526499125662, 5.37087692657741975151146833972, 5.50809492341416810523843782690, 5.53180956507558771160557584707, 5.59492169770929319356041996713, 5.79921657306849615897391611616, 6.15801425521371923824834134501, 6.28699124025639060076281459193
