| L(s) = 1 | − 2·5-s − 9-s + 8·11-s − 25-s − 12·29-s + 8·31-s − 20·41-s + 2·45-s − 2·49-s − 16·55-s + 8·59-s − 4·61-s + 24·79-s + 81-s + 20·89-s − 8·99-s + 4·101-s − 4·109-s + 26·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1/3·9-s + 2.41·11-s − 1/5·25-s − 2.22·29-s + 1.43·31-s − 3.12·41-s + 0.298·45-s − 2/7·49-s − 2.15·55-s + 1.04·59-s − 0.512·61-s + 2.70·79-s + 1/9·81-s + 2.11·89-s − 0.804·99-s + 0.398·101-s − 0.383·109-s + 2.36·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.574036456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.574036456\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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
−10.30276028311646284696256212704, −9.604741249989206962600721807586, −9.465728689451772004104393721753, −8.928141589998657826131304667301, −8.654180746930419273281479283479, −8.174594050522944957499214630896, −7.71233027625070024291179211903, −7.34975449348103629870469501371, −6.71798277361060597498288699117, −6.42145948028343112287635055282, −6.23131352613987341799417887419, −5.21530105848065634863351575877, −5.17377208552795594871462447013, −4.21285046948558435800465299820, −4.00957946078956219075303039559, −3.49579210520965279779946907598, −3.17123838125889381459660932337, −2.03467005914230254452812125459, −1.60287055768205907766400981082, −0.60663786219540718786806093953,
0.60663786219540718786806093953, 1.60287055768205907766400981082, 2.03467005914230254452812125459, 3.17123838125889381459660932337, 3.49579210520965279779946907598, 4.00957946078956219075303039559, 4.21285046948558435800465299820, 5.17377208552795594871462447013, 5.21530105848065634863351575877, 6.23131352613987341799417887419, 6.42145948028343112287635055282, 6.71798277361060597498288699117, 7.34975449348103629870469501371, 7.71233027625070024291179211903, 8.174594050522944957499214630896, 8.654180746930419273281479283479, 8.928141589998657826131304667301, 9.465728689451772004104393721753, 9.604741249989206962600721807586, 10.30276028311646284696256212704
