| L(s) = 1 | + 8·11-s − 12·13-s − 12·23-s + 10·25-s − 12·37-s + 12·47-s − 4·49-s − 8·59-s + 12·61-s + 12·71-s + 12·73-s + 32·83-s − 24·97-s + 32·107-s + 12·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 96·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + ⋯ |
| L(s) = 1 | + 2.41·11-s − 3.32·13-s − 2.50·23-s + 2·25-s − 1.97·37-s + 1.75·47-s − 4/7·49-s − 1.04·59-s + 1.53·61-s + 1.42·71-s + 1.40·73-s + 3.51·83-s − 2.43·97-s + 3.09·107-s + 1.14·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 8.02·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 6.30·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.913015216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913015216\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.379242369129545391056555770967, −8.281727875996340205926728840513, −7.54836601528391915874341346443, −7.47492027985152493306872111186, −7.00541818793748716783120293566, −6.69741620459850175635473403581, −6.50033308898321330160115165509, −6.05531711658694102965216739370, −5.43456788876945991080903544483, −5.18256315039191513160615503854, −4.67740197380450760806122119072, −4.53845935531958948573654704694, −3.95998620672111673457320435944, −3.59157795495890434429487960703, −3.24675181572872360970706542169, −2.45500455513515700810367104301, −2.15372077953322848092943999975, −1.88672077314071183445224226437, −1.03182485936910985551663084346, −0.42060422220265417722496646386,
0.42060422220265417722496646386, 1.03182485936910985551663084346, 1.88672077314071183445224226437, 2.15372077953322848092943999975, 2.45500455513515700810367104301, 3.24675181572872360970706542169, 3.59157795495890434429487960703, 3.95998620672111673457320435944, 4.53845935531958948573654704694, 4.67740197380450760806122119072, 5.18256315039191513160615503854, 5.43456788876945991080903544483, 6.05531711658694102965216739370, 6.50033308898321330160115165509, 6.69741620459850175635473403581, 7.00541818793748716783120293566, 7.47492027985152493306872111186, 7.54836601528391915874341346443, 8.281727875996340205926728840513, 8.379242369129545391056555770967
