| 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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.328205216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328205216\) |
| \(L(3)\) |
|
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
−12.84916597948321019744020658031, −11.46306630083919058799991005559, −11.40333927377325053069789760881, −10.50291846939009242301056616604, −10.10102355314160471155465215291, −9.669645848157349336931653248894, −9.333398730970440390461360051757, −8.863298479168645485062363662466, −8.359390507847896322245139700251, −7.58711368502430422908492726568, −6.99736222701741562497830389145, −6.25444151730614711101810693560, −5.78548789749374492836152113568, −5.16954847113857117491254903955, −4.97453756790417294903026786853, −3.79481799917123936486687059950, −3.33081549800249446497849767422, −2.06880317026465079436632399010, −1.73930764885591424154845913215, −0.56472551459403432736116166542,
0.56472551459403432736116166542, 1.73930764885591424154845913215, 2.06880317026465079436632399010, 3.33081549800249446497849767422, 3.79481799917123936486687059950, 4.97453756790417294903026786853, 5.16954847113857117491254903955, 5.78548789749374492836152113568, 6.25444151730614711101810693560, 6.99736222701741562497830389145, 7.58711368502430422908492726568, 8.359390507847896322245139700251, 8.863298479168645485062363662466, 9.333398730970440390461360051757, 9.669645848157349336931653248894, 10.10102355314160471155465215291, 10.50291846939009242301056616604, 11.40333927377325053069789760881, 11.46306630083919058799991005559, 12.84916597948321019744020658031
