| L(s) = 1 | + 5-s + 2·7-s − 3·11-s − 4·17-s + 2·19-s + 2·23-s − 3·29-s + 3·31-s + 2·35-s + 5·41-s + 4·43-s − 8·47-s + 7·49-s − 4·53-s − 3·55-s + 3·59-s − 6·61-s + 10·67-s + 30·71-s − 28·73-s − 6·77-s + 8·79-s − 4·85-s − 2·89-s + 2·95-s + 16·97-s − 13·101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.904·11-s − 0.970·17-s + 0.458·19-s + 0.417·23-s − 0.557·29-s + 0.538·31-s + 0.338·35-s + 0.780·41-s + 0.609·43-s − 1.16·47-s + 49-s − 0.549·53-s − 0.404·55-s + 0.390·59-s − 0.768·61-s + 1.22·67-s + 3.56·71-s − 3.27·73-s − 0.683·77-s + 0.900·79-s − 0.433·85-s − 0.211·89-s + 0.205·95-s + 1.62·97-s − 1.29·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.493687040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.493687040\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 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
−8.859383177348648961639434785688, −8.390211810370216745053869453037, −8.020686824011501941423173182187, −7.901953392606441903156166470790, −7.34134278297474276721336431124, −6.95288576387424784941635844134, −6.72010435543885883497114640850, −6.10116272362938215869034109214, −5.79535061399414136883985028713, −5.44315924915665834002457519154, −4.90568744544111468928161662043, −4.77678298367251216081951370752, −4.25400638760413481746975810689, −3.77347468516590845171401239498, −3.21800101835331387635533616547, −2.71626551976141000680750369473, −2.27076516114054199582554604550, −1.89055452357118138688555017607, −1.21081840725337483660823983582, −0.50497182449058796203098959131,
0.50497182449058796203098959131, 1.21081840725337483660823983582, 1.89055452357118138688555017607, 2.27076516114054199582554604550, 2.71626551976141000680750369473, 3.21800101835331387635533616547, 3.77347468516590845171401239498, 4.25400638760413481746975810689, 4.77678298367251216081951370752, 4.90568744544111468928161662043, 5.44315924915665834002457519154, 5.79535061399414136883985028713, 6.10116272362938215869034109214, 6.72010435543885883497114640850, 6.95288576387424784941635844134, 7.34134278297474276721336431124, 7.901953392606441903156166470790, 8.020686824011501941423173182187, 8.390211810370216745053869453037, 8.859383177348648961639434785688
