| L(s) = 1 | − 4·5-s + 11·25-s − 8·29-s + 16·41-s + 14·49-s + 20·61-s + 32·89-s − 40·101-s + 12·109-s − 22·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 11/5·25-s − 1.48·29-s + 2.49·41-s + 2·49-s + 2.56·61-s + 3.39·89-s − 3.98·101-s + 1.14·109-s − 2·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.183443563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.183443563\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.458679988768253084379139758445, −9.357567053701169161926616503547, −8.925301369306121627365714418917, −8.453381622514453753375831803736, −8.050081986763397716150557060162, −7.73887529773438920537607144060, −7.31092479107903217698039894091, −7.14792364725153639479603561366, −6.54525525749204763048290778765, −6.09046397723231810391365551442, −5.41502213200377348833706332231, −5.26633385497965809184663136883, −4.49710116028780179666916488972, −4.08541305420778551649624752280, −3.83470254322702316279446150214, −3.41091975851096070929084056928, −2.61159662032030680773899388489, −2.27971455438168325708404875579, −1.15705028952187128664649683041, −0.50473177986945510754361832342,
0.50473177986945510754361832342, 1.15705028952187128664649683041, 2.27971455438168325708404875579, 2.61159662032030680773899388489, 3.41091975851096070929084056928, 3.83470254322702316279446150214, 4.08541305420778551649624752280, 4.49710116028780179666916488972, 5.26633385497965809184663136883, 5.41502213200377348833706332231, 6.09046397723231810391365551442, 6.54525525749204763048290778765, 7.14792364725153639479603561366, 7.31092479107903217698039894091, 7.73887529773438920537607144060, 8.050081986763397716150557060162, 8.453381622514453753375831803736, 8.925301369306121627365714418917, 9.357567053701169161926616503547, 9.458679988768253084379139758445
