| L(s) = 1 | + 2·3-s − 2·4-s + 9-s − 4·12-s + 4·16-s + 12·19-s + 6·25-s − 4·27-s − 2·36-s − 4·43-s + 8·48-s − 49-s + 24·57-s − 8·64-s − 4·67-s − 12·73-s + 12·75-s − 24·76-s − 11·81-s − 4·97-s − 12·100-s + 8·108-s − 18·121-s + 127-s − 8·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s + 1/3·9-s − 1.15·12-s + 16-s + 2.75·19-s + 6/5·25-s − 0.769·27-s − 1/3·36-s − 0.609·43-s + 1.15·48-s − 1/7·49-s + 3.17·57-s − 64-s − 0.488·67-s − 1.40·73-s + 1.38·75-s − 2.75·76-s − 1.22·81-s − 0.406·97-s − 6/5·100-s + 0.769·108-s − 1.63·121-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472718581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472718581\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−10.28620399341622470550638453054, −9.887136630325027490666649135811, −9.393507695232222751319006863113, −9.002136739186055216164212439106, −8.567329421530738796335254021138, −7.938646266925342778705410318895, −7.53208109086882074354712245319, −6.98762414182075845869423032524, −5.99375492422382120737687149168, −5.31306434220500476035379994936, −4.87550560097527312407599615644, −3.96019251490424761722410398205, −3.27001075285133692901361970260, −2.82526965718183128765437634941, −1.30990847497478007676972301682,
1.30990847497478007676972301682, 2.82526965718183128765437634941, 3.27001075285133692901361970260, 3.96019251490424761722410398205, 4.87550560097527312407599615644, 5.31306434220500476035379994936, 5.99375492422382120737687149168, 6.98762414182075845869423032524, 7.53208109086882074354712245319, 7.938646266925342778705410318895, 8.567329421530738796335254021138, 9.002136739186055216164212439106, 9.393507695232222751319006863113, 9.887136630325027490666649135811, 10.28620399341622470550638453054
