| L(s) = 1 | − 2·3-s − 4·7-s + 9-s + 4·19-s + 8·21-s + 2·25-s + 4·27-s + 16·31-s + 4·37-s + 9·49-s − 8·57-s − 4·63-s − 4·75-s − 11·81-s − 32·93-s − 8·103-s + 28·109-s − 8·111-s − 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s − 18·147-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.917·19-s + 1.74·21-s + 2/5·25-s + 0.769·27-s + 2.87·31-s + 0.657·37-s + 9/7·49-s − 1.05·57-s − 0.503·63-s − 0.461·75-s − 1.22·81-s − 3.31·93-s − 0.788·103-s + 2.68·109-s − 0.759·111-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-s + 0.0854·137-s + 0.0848·139-s − 1.48·147-s + 0.0819·149-s + 0.0813·151-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
| \(L(1)\) |
\(\approx\) |
\(0.6451089248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6451089248\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−10.42175455305646155037529200631, −10.13598371854604000779538496815, −9.697048900588129159041990401969, −9.123435915447189262623776904610, −8.471591447824925262966853131863, −7.83392461629794701403201815756, −7.07345093709724690071287703176, −6.54305365458703917515509937391, −6.20409897661964783063656261325, −5.63818606439032529206874539044, −4.91682693429455677330737022363, −4.28215112882699580202983901503, −3.24929906692181964024640809239, −2.70549066195282146083432768330, −0.837590667334905948714056378019,
0.837590667334905948714056378019, 2.70549066195282146083432768330, 3.24929906692181964024640809239, 4.28215112882699580202983901503, 4.91682693429455677330737022363, 5.63818606439032529206874539044, 6.20409897661964783063656261325, 6.54305365458703917515509937391, 7.07345093709724690071287703176, 7.83392461629794701403201815756, 8.471591447824925262966853131863, 9.123435915447189262623776904610, 9.697048900588129159041990401969, 10.13598371854604000779538496815, 10.42175455305646155037529200631
