| L(s) = 1 | − 4·4-s − 5-s − 6·9-s + 5·11-s + 12·16-s + 4·20-s − 4·25-s + 24·36-s − 20·44-s + 6·45-s + 5·49-s − 5·55-s − 32·64-s − 12·80-s + 27·81-s − 30·99-s + 16·100-s + 14·121-s + 9·125-s + 127-s + 131-s + 137-s + 139-s − 72·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2·4-s − 0.447·5-s − 2·9-s + 1.50·11-s + 3·16-s + 0.894·20-s − 4/5·25-s + 4·36-s − 3.01·44-s + 0.894·45-s + 5/7·49-s − 0.674·55-s − 4·64-s − 1.34·80-s + 3·81-s − 3.01·99-s + 8/5·100-s + 1.27·121-s + 0.804·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5432302084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5432302084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−8.257438293285400064487372330011, −7.80721810924236528989148696773, −7.42576936308490743919691601661, −6.60574857586478393452876072930, −6.22062063563158446237036032778, −5.67646921885699346859765144678, −5.46158387192313632246151765332, −4.91092734270594761008678887775, −4.27523868766142284236019982561, −4.05860439288340487411118779986, −3.39451292921803621175036861493, −3.20621111912967275630179366480, −2.24255268758052633749339242499, −1.19465266884302745044451365528, −0.39402131572052920383039003444,
0.39402131572052920383039003444, 1.19465266884302745044451365528, 2.24255268758052633749339242499, 3.20621111912967275630179366480, 3.39451292921803621175036861493, 4.05860439288340487411118779986, 4.27523868766142284236019982561, 4.91092734270594761008678887775, 5.46158387192313632246151765332, 5.67646921885699346859765144678, 6.22062063563158446237036032778, 6.60574857586478393452876072930, 7.42576936308490743919691601661, 7.80721810924236528989148696773, 8.257438293285400064487372330011
