L(s) = 1 | − 9-s − 12·11-s + 8·19-s − 12·29-s + 16·31-s + 24·41-s − 49-s − 20·61-s + 12·71-s + 8·79-s + 81-s − 24·89-s + 12·99-s − 24·101-s − 28·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 8·171-s + ⋯ |
L(s) = 1 | − 1/3·9-s − 3.61·11-s + 1.83·19-s − 2.22·29-s + 2.87·31-s + 3.74·41-s − 1/7·49-s − 2.56·61-s + 1.42·71-s + 0.900·79-s + 1/9·81-s − 2.54·89-s + 1.20·99-s − 2.38·101-s − 2.68·109-s + 7.81·121-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 + 1.69·169-s − 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.142303540\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.142303540\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.334160588088703665723149310247, −9.056382081793747333227466982444, −8.112979180081501232426812543219, −8.096362834242939938755733891477, −7.84588758143579943271053413032, −7.58239115349956445927609861427, −7.09959069637711725499202161259, −6.62069975807823456338094688079, −5.79532631877383755956047939166, −5.61235911729251425840469039932, −5.56894161311508485763590873490, −4.94175526113973912105672302566, −4.50078541102630895590818772348, −4.13766361371917066456946450974, −3.07919939410614854098454602045, −3.07240320708764163054184539503, −2.56113527066698518566206620410, −2.20418094039544649709065316617, −1.16120108529434125444805543616, −0.40506453482796761740976323754,
0.40506453482796761740976323754, 1.16120108529434125444805543616, 2.20418094039544649709065316617, 2.56113527066698518566206620410, 3.07240320708764163054184539503, 3.07919939410614854098454602045, 4.13766361371917066456946450974, 4.50078541102630895590818772348, 4.94175526113973912105672302566, 5.56894161311508485763590873490, 5.61235911729251425840469039932, 5.79532631877383755956047939166, 6.62069975807823456338094688079, 7.09959069637711725499202161259, 7.58239115349956445927609861427, 7.84588758143579943271053413032, 8.096362834242939938755733891477, 8.112979180081501232426812543219, 9.056382081793747333227466982444, 9.334160588088703665723149310247