| L(s) = 1 | − 2·4-s − 5·16-s + 10·25-s + 8·43-s + 4·49-s − 28·61-s + 20·64-s + 32·79-s − 20·100-s + 8·103-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 8·196-s + ⋯ |
| L(s) = 1 | − 4-s − 5/4·16-s + 2·25-s + 1.21·43-s + 4/7·49-s − 3.58·61-s + 5/2·64-s + 3.60·79-s − 2·100-s + 0.788·103-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.21·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4/7·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.375953782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.375953782\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 43 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 71 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 77 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 29 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.64159977389886483374733543356, −6.57192372761666862511521532854, −6.44855959734804746737196837144, −6.09862531398193875059330931586, −5.77703494990534050080790329861, −5.68544135187376117286158592520, −5.54106704023689582266312675880, −4.99425969732689392131662905330, −4.91064363271463126368566293024, −4.72646366422365804226531246558, −4.56040796102787722509330402569, −4.54243728070389386923773882543, −4.07196669363905653923415204225, −3.78320665418669620586586688016, −3.68819759954908036616699835834, −3.29831161749230545472191696962, −2.97206013539901629173216003610, −2.93610114725805952347804256971, −2.34164859450647358405726256258, −2.27023542560676347018732155092, −2.04706693357494641539220347850, −1.44153397907983637862654706229, −1.15006919362651788259404632993, −0.73785669941696857544253395231, −0.29775868080061585941786165749,
0.29775868080061585941786165749, 0.73785669941696857544253395231, 1.15006919362651788259404632993, 1.44153397907983637862654706229, 2.04706693357494641539220347850, 2.27023542560676347018732155092, 2.34164859450647358405726256258, 2.93610114725805952347804256971, 2.97206013539901629173216003610, 3.29831161749230545472191696962, 3.68819759954908036616699835834, 3.78320665418669620586586688016, 4.07196669363905653923415204225, 4.54243728070389386923773882543, 4.56040796102787722509330402569, 4.72646366422365804226531246558, 4.91064363271463126368566293024, 4.99425969732689392131662905330, 5.54106704023689582266312675880, 5.68544135187376117286158592520, 5.77703494990534050080790329861, 6.09862531398193875059330931586, 6.44855959734804746737196837144, 6.57192372761666862511521532854, 6.64159977389886483374733543356
