| L(s) = 1 | + 6·9-s + 4·17-s + 20·41-s − 14·49-s − 12·73-s + 27·81-s − 20·89-s + 36·97-s + 28·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2·9-s + 0.970·17-s + 3.12·41-s − 2·49-s − 1.40·73-s + 3·81-s − 2.11·89-s + 3.65·97-s + 2.63·113-s + 2·121-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 + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.889192411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.889192411\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.771621238555295582313835371606, −8.553248814621543414559476563048, −7.962748886379707596434734267393, −7.62990945226927632845305968034, −7.32439499380261798354075326981, −7.23893537670033320295523467281, −6.57804545545840312001509132333, −6.26233683457792968630773813863, −5.82520886959755525616969532327, −5.54393653160422277646327294623, −4.71942789328790503606459580393, −4.69748299905297721054346964029, −4.24092513712175700102627758335, −3.84037559500478732364104226705, −3.21659745691824248076399696267, −3.00619302827288649471824217852, −2.07621330713333084387651272421, −1.85469028369450052785797693339, −1.11944134244234932461499603278, −0.70883904967130064489406282429,
0.70883904967130064489406282429, 1.11944134244234932461499603278, 1.85469028369450052785797693339, 2.07621330713333084387651272421, 3.00619302827288649471824217852, 3.21659745691824248076399696267, 3.84037559500478732364104226705, 4.24092513712175700102627758335, 4.69748299905297721054346964029, 4.71942789328790503606459580393, 5.54393653160422277646327294623, 5.82520886959755525616969532327, 6.26233683457792968630773813863, 6.57804545545840312001509132333, 7.23893537670033320295523467281, 7.32439499380261798354075326981, 7.62990945226927632845305968034, 7.962748886379707596434734267393, 8.553248814621543414559476563048, 8.771621238555295582313835371606
