| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s + 8·10-s + 2·11-s − 4·16-s + 4·17-s − 4·19-s − 8·20-s − 4·22-s + 8·25-s + 14·29-s + 16·31-s + 8·32-s − 8·34-s − 10·37-s + 8·38-s − 2·43-s + 4·44-s + 24·47-s − 16·50-s − 2·53-s − 8·55-s − 28·58-s − 16·59-s − 12·61-s − 32·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s + 2.52·10-s + 0.603·11-s − 16-s + 0.970·17-s − 0.917·19-s − 1.78·20-s − 0.852·22-s + 8/5·25-s + 2.59·29-s + 2.87·31-s + 1.41·32-s − 1.37·34-s − 1.64·37-s + 1.29·38-s − 0.304·43-s + 0.603·44-s + 3.50·47-s − 2.26·50-s − 0.274·53-s − 1.07·55-s − 3.67·58-s − 2.08·59-s − 1.53·61-s − 4.06·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6620078399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6620078399\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.44111012520458151237336616860, −10.25659233632624403058140503210, −9.558580908218755588735998872242, −9.136509592257109452326992971534, −8.729460618470114343075716107426, −8.292922770104392256645978453947, −7.971842557862310803922103731323, −7.897046213596400062573097288406, −7.02703721455337747646231827117, −6.96382996137667195536448325336, −6.34127366071256277371655773187, −5.91864304319347871997126854932, −4.83270976941007915784579483180, −4.57010123587605586027017564728, −4.22729191304944437007389581291, −3.45090509881951208464321337002, −2.97820188139708549960425720950, −2.21925090854364058198112603049, −1.10861372974426628568882498339, −0.64909663662900655600071297757,
0.64909663662900655600071297757, 1.10861372974426628568882498339, 2.21925090854364058198112603049, 2.97820188139708549960425720950, 3.45090509881951208464321337002, 4.22729191304944437007389581291, 4.57010123587605586027017564728, 4.83270976941007915784579483180, 5.91864304319347871997126854932, 6.34127366071256277371655773187, 6.96382996137667195536448325336, 7.02703721455337747646231827117, 7.897046213596400062573097288406, 7.971842557862310803922103731323, 8.292922770104392256645978453947, 8.729460618470114343075716107426, 9.136509592257109452326992971534, 9.558580908218755588735998872242, 10.25659233632624403058140503210, 10.44111012520458151237336616860
