| L(s) = 1 | − 5-s + 9-s + 8·13-s + 25-s + 4·31-s − 4·37-s − 8·41-s + 12·43-s − 45-s − 2·49-s + 4·53-s − 8·65-s + 16·67-s − 12·71-s + 12·79-s + 81-s + 4·83-s − 8·89-s + 4·107-s + 8·117-s − 10·121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1/3·9-s + 2.21·13-s + 1/5·25-s + 0.718·31-s − 0.657·37-s − 1.24·41-s + 1.82·43-s − 0.149·45-s − 2/7·49-s + 0.549·53-s − 0.992·65-s + 1.95·67-s − 1.42·71-s + 1.35·79-s + 1/9·81-s + 0.439·83-s − 0.847·89-s + 0.386·107-s + 0.739·117-s − 0.909·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.888876316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.888876316\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−8.771816694199603173075710856479, −8.437178202393953326891601687037, −7.993494115960680951848118050841, −7.52146879103278953106650325424, −6.89603544660867909998908619024, −6.51025722695981470829358788244, −6.05906112392509166729158907540, −5.51972098412047532803301062654, −4.94251524859779437951685647069, −4.25720585341674706911017391327, −3.78865602892148938984040920376, −3.40750142549406989097737265031, −2.59319807407447705771899584414, −1.64982095045562812587562733390, −0.882736108663337361901498306377,
0.882736108663337361901498306377, 1.64982095045562812587562733390, 2.59319807407447705771899584414, 3.40750142549406989097737265031, 3.78865602892148938984040920376, 4.25720585341674706911017391327, 4.94251524859779437951685647069, 5.51972098412047532803301062654, 6.05906112392509166729158907540, 6.51025722695981470829358788244, 6.89603544660867909998908619024, 7.52146879103278953106650325424, 7.993494115960680951848118050841, 8.437178202393953326891601687037, 8.771816694199603173075710856479
