| L(s) = 1 | − 16·4-s − 48·5-s + 256·16-s + 768·20-s + 1.67e3·25-s + 3.36e3·29-s + 2.88e3·41-s − 4.80e3·49-s + 1.39e4·61-s − 4.09e3·64-s − 1.22e4·80-s − 2.49e4·89-s − 2.68e4·100-s − 1.58e4·101-s + 1.87e4·109-s − 5.37e4·116-s + 2.92e4·121-s − 5.05e4·125-s + 127-s + 131-s + 137-s + 139-s − 1.61e5·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s − 1.91·5-s + 16-s + 1.91·20-s + 2.68·25-s + 3.99·29-s + 1.71·41-s − 2·49-s + 3.73·61-s − 64-s − 1.91·80-s − 3.15·89-s − 2.68·100-s − 1.55·101-s + 1.57·109-s − 3.99·116-s + 2·121-s − 3.23·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 7.67·145-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.202727420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.202727420\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 48 T + p^{4} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 238 T + p^{4} T^{2} )( 1 + 238 T + p^{4} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 480 T + p^{4} T^{2} )( 1 + 480 T + p^{4} T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 1680 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2162 T + p^{4} T^{2} )( 1 + 2162 T + p^{4} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 1440 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 5040 T + p^{4} T^{2} )( 1 + 5040 T + p^{4} T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6958 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1442 T + p^{4} T^{2} )( 1 + 1442 T + p^{4} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12480 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 1918 T + p^{4} T^{2} )( 1 + 1918 T + p^{4} T^{2} ) \) |
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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
−12.32867050089341339014993740881, −11.76280325022573100541090202018, −11.29642614266530509700145249822, −10.88524060647061758749970577951, −10.00059361524835441391902258885, −9.993971361846747435251102097128, −9.023030944777648961712662586907, −8.496254868576611028714002955756, −8.135595067949400217009264917798, −7.973346253792331643831472716640, −6.88197965118615478422482852383, −6.80893568359234311941556282622, −5.75313572382109520343230615062, −4.94391859132391808072310724856, −4.45789060864983485351110363819, −4.12513983654830930782600294087, −3.27583880921290937872745315967, −2.74898501211012022883929653840, −1.01213107247189819569353198068, −0.54361675723915932069824384696,
0.54361675723915932069824384696, 1.01213107247189819569353198068, 2.74898501211012022883929653840, 3.27583880921290937872745315967, 4.12513983654830930782600294087, 4.45789060864983485351110363819, 4.94391859132391808072310724856, 5.75313572382109520343230615062, 6.80893568359234311941556282622, 6.88197965118615478422482852383, 7.973346253792331643831472716640, 8.135595067949400217009264917798, 8.496254868576611028714002955756, 9.023030944777648961712662586907, 9.993971361846747435251102097128, 10.00059361524835441391902258885, 10.88524060647061758749970577951, 11.29642614266530509700145249822, 11.76280325022573100541090202018, 12.32867050089341339014993740881
