| L(s) = 1 | − 3·4-s − 7-s − 2·9-s + 11-s + 8·13-s + 5·16-s + 8·17-s − 8·23-s − 6·25-s + 3·28-s + 6·36-s − 12·37-s + 8·41-s − 3·44-s + 49-s − 24·52-s − 12·53-s + 2·63-s − 3·64-s + 16·67-s − 24·68-s − 24·71-s − 16·73-s − 77-s − 5·81-s − 8·91-s + 24·92-s + ⋯ |
| L(s) = 1 | − 3/2·4-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 2.21·13-s + 5/4·16-s + 1.94·17-s − 1.66·23-s − 6/5·25-s + 0.566·28-s + 36-s − 1.97·37-s + 1.24·41-s − 0.452·44-s + 1/7·49-s − 3.32·52-s − 1.64·53-s + 0.251·63-s − 3/8·64-s + 1.95·67-s − 2.91·68-s − 2.84·71-s − 1.87·73-s − 0.113·77-s − 5/9·81-s − 0.838·91-s + 2.50·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456533 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456533 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.583198427577347970514937919978, −7.82867506007281568614416512666, −7.82735023630063825425322329351, −6.99177048985268727820100186274, −6.13470624546121738215110863867, −5.96443978110022033983848919969, −5.66505626598251892142666651623, −5.11723556108121000911865118811, −4.18761328115181872919457156606, −4.07930883034575373640745113861, −3.29721011206473421957461335105, −3.26888344602981049454952325554, −1.84332651626064700387015155391, −1.12875699652193550593288105654, 0,
1.12875699652193550593288105654, 1.84332651626064700387015155391, 3.26888344602981049454952325554, 3.29721011206473421957461335105, 4.07930883034575373640745113861, 4.18761328115181872919457156606, 5.11723556108121000911865118811, 5.66505626598251892142666651623, 5.96443978110022033983848919969, 6.13470624546121738215110863867, 6.99177048985268727820100186274, 7.82735023630063825425322329351, 7.82867506007281568614416512666, 8.583198427577347970514937919978
