L(s) = 1 | + 4·5-s + 12·13-s − 4·17-s + 2·25-s + 20·29-s − 4·37-s − 20·41-s − 14·49-s − 28·53-s − 20·61-s + 48·65-s − 12·73-s − 16·85-s − 20·89-s + 36·97-s + 4·101-s + 12·109-s + 28·113-s − 22·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 80·145-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 3.32·13-s − 0.970·17-s + 2/5·25-s + 3.71·29-s − 0.657·37-s − 3.12·41-s − 2·49-s − 3.84·53-s − 2.56·61-s + 5.95·65-s − 1.40·73-s − 1.73·85-s − 2.11·89-s + 3.65·97-s + 0.398·101-s + 1.14·109-s + 2.63·113-s − 2·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 6.64·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.291728606\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.291728606\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−14.17235133347833968439098790748, −13.64951405560498208286110774441, −13.64951405560498208286110774441, −12.89009592608142214221474186002, −12.89009592608142214221474186002, −11.74104183190432762188729414275, −11.74104183190432762188729414275, −10.79424580459691548950831956254, −10.79424580459691548950831956254, −9.988093934240586032989036843847, −9.988093934240586032989036843847, −8.944850116469648867972974935175, −8.944850116469648867972974935175, −8.155593376401658883389001410970, −8.155593376401658883389001410970, −6.62912355829218159331214255175, −6.62912355829218159331214255175, −5.98788019309426758968327152208, −5.98788019309426758968327152208, −4.69735185642958092576120623346, −4.69735185642958092576120623346, −3.23795791772265289886094041940, −3.23795791772265289886094041940, −1.61821392697617915675557729976, −1.61821392697617915675557729976,
1.61821392697617915675557729976, 1.61821392697617915675557729976, 3.23795791772265289886094041940, 3.23795791772265289886094041940, 4.69735185642958092576120623346, 4.69735185642958092576120623346, 5.98788019309426758968327152208, 5.98788019309426758968327152208, 6.62912355829218159331214255175, 6.62912355829218159331214255175, 8.155593376401658883389001410970, 8.155593376401658883389001410970, 8.944850116469648867972974935175, 8.944850116469648867972974935175, 9.988093934240586032989036843847, 9.988093934240586032989036843847, 10.79424580459691548950831956254, 10.79424580459691548950831956254, 11.74104183190432762188729414275, 11.74104183190432762188729414275, 12.89009592608142214221474186002, 12.89009592608142214221474186002, 13.64951405560498208286110774441, 13.64951405560498208286110774441, 14.17235133347833968439098790748