L(s) = 1 | − 3·3-s + 5-s − 5·7-s + 6·9-s + 2·11-s + 6·13-s − 3·15-s + 16·17-s + 8·19-s + 15·21-s − 3·23-s − 9·27-s − 7·29-s + 4·31-s − 6·33-s − 5·35-s − 8·37-s − 18·39-s − 5·41-s − 8·43-s + 6·45-s + 47-s + 7·49-s − 48·51-s + 4·53-s + 2·55-s − 24·57-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 0.447·5-s − 1.88·7-s + 2·9-s + 0.603·11-s + 1.66·13-s − 0.774·15-s + 3.88·17-s + 1.83·19-s + 3.27·21-s − 0.625·23-s − 1.73·27-s − 1.29·29-s + 0.718·31-s − 1.04·33-s − 0.845·35-s − 1.31·37-s − 2.88·39-s − 0.780·41-s − 1.21·43-s + 0.894·45-s + 0.145·47-s + 49-s − 6.72·51-s + 0.549·53-s + 0.269·55-s − 3.17·57-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
\(L(1)\) |
\(\approx\) |
\(1.260667588\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.260667588\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.788675505065638279740169601691, −9.646434563821847536543816398256, −9.180383913935014064714402943114, −8.569101179506431051513367208375, −7.976565115662929613618817037903, −7.63188340315851142799106847976, −7.09828196050498414289953152241, −6.77929373783523351897564452638, −6.34092126675569775696386139256, −5.86896892793052493613509974411, −5.77722611040151075121486977705, −5.29551525967771654862793329106, −5.14198440652983944015443441375, −4.01620889716993095072539899835, −3.67346766278826708870417559443, −3.27060311915371015399232914345, −3.08091811530681106097017203688, −1.48088463375938448566969119497, −1.35816500672471524511137407558, −0.58699944297423921081919902797,
0.58699944297423921081919902797, 1.35816500672471524511137407558, 1.48088463375938448566969119497, 3.08091811530681106097017203688, 3.27060311915371015399232914345, 3.67346766278826708870417559443, 4.01620889716993095072539899835, 5.14198440652983944015443441375, 5.29551525967771654862793329106, 5.77722611040151075121486977705, 5.86896892793052493613509974411, 6.34092126675569775696386139256, 6.77929373783523351897564452638, 7.09828196050498414289953152241, 7.63188340315851142799106847976, 7.976565115662929613618817037903, 8.569101179506431051513367208375, 9.180383913935014064714402943114, 9.646434563821847536543816398256, 9.788675505065638279740169601691