| L(s) = 1 | − 3-s + 9-s + 8·11-s + 6·25-s − 27-s − 8·33-s − 2·49-s + 8·59-s + 12·73-s − 6·75-s + 81-s + 24·83-s − 28·97-s + 8·99-s + 8·107-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 2·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 2.41·11-s + 6/5·25-s − 0.192·27-s − 1.39·33-s − 2/7·49-s + 1.04·59-s + 1.40·73-s − 0.692·75-s + 1/9·81-s + 2.63·83-s − 2.84·97-s + 0.804·99-s + 0.773·107-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.164·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 442368 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.914981930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.914981930\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 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
−8.684043979652876988066025176599, −8.198948572611240427865165792362, −7.61532598445505021231566482492, −6.98089040034932339267148531426, −6.71372099189548872370259188607, −6.39335751084534575278078850353, −5.91040934760801511512508817965, −5.21950175494313500320865809431, −4.83814835762951883923045406987, −4.13901961167284522711150074983, −3.82169131902864432935982281522, −3.23290331977889648543182041186, −2.32652027208272497884774882902, −1.48465297543568880446390570490, −0.870220222855083091023826253514,
0.870220222855083091023826253514, 1.48465297543568880446390570490, 2.32652027208272497884774882902, 3.23290331977889648543182041186, 3.82169131902864432935982281522, 4.13901961167284522711150074983, 4.83814835762951883923045406987, 5.21950175494313500320865809431, 5.91040934760801511512508817965, 6.39335751084534575278078850353, 6.71372099189548872370259188607, 6.98089040034932339267148531426, 7.61532598445505021231566482492, 8.198948572611240427865165792362, 8.684043979652876988066025176599
