| L(s) = 1 | − 3·5-s − 3·11-s − 4·13-s − 3·17-s − 19-s + 3·23-s + 5·25-s + 12·29-s − 7·31-s + 37-s + 12·41-s − 8·43-s + 9·47-s + 3·53-s + 9·55-s − 9·59-s − 61-s + 12·65-s + 7·67-s − 73-s + 13·79-s + 24·83-s + 9·85-s − 15·89-s + 3·95-s + 20·97-s − 15·101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.904·11-s − 1.10·13-s − 0.727·17-s − 0.229·19-s + 0.625·23-s + 25-s + 2.22·29-s − 1.25·31-s + 0.164·37-s + 1.87·41-s − 1.21·43-s + 1.31·47-s + 0.412·53-s + 1.21·55-s − 1.17·59-s − 0.128·61-s + 1.48·65-s + 0.855·67-s − 0.117·73-s + 1.46·79-s + 2.63·83-s + 0.976·85-s − 1.58·89-s + 0.307·95-s + 2.03·97-s − 1.49·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8187836127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8187836127\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.275993192494616111285028642311, −9.126592344308872936867435904246, −8.687065203536388825834989353791, −8.231753682213813063892496311277, −7.88491011632925528730026328762, −7.52057771591072305014124119561, −7.26473740922889085785367969412, −6.86130523218797883858007644754, −6.24281146049264502037391873351, −6.06474461249329583708042060231, −5.04289591535035546902652983735, −5.03699017945427321285348744946, −4.69151708123459202889501143222, −4.02173828309510198941301179338, −3.70995277526017633003680313193, −3.08129912057261977144558726871, −2.42099977521082369946778193307, −2.37441098363492605969234423041, −1.13677278297615577680817401978, −0.37839860505665914040816438767,
0.37839860505665914040816438767, 1.13677278297615577680817401978, 2.37441098363492605969234423041, 2.42099977521082369946778193307, 3.08129912057261977144558726871, 3.70995277526017633003680313193, 4.02173828309510198941301179338, 4.69151708123459202889501143222, 5.03699017945427321285348744946, 5.04289591535035546902652983735, 6.06474461249329583708042060231, 6.24281146049264502037391873351, 6.86130523218797883858007644754, 7.26473740922889085785367969412, 7.52057771591072305014124119561, 7.88491011632925528730026328762, 8.231753682213813063892496311277, 8.687065203536388825834989353791, 9.126592344308872936867435904246, 9.275993192494616111285028642311
