| L(s) = 1 | − 4·5-s + 3·7-s − 4·11-s − 13-s − 8·17-s − 2·19-s − 4·23-s + 5·25-s + 4·31-s − 12·35-s − 18·37-s + 8·43-s + 12·47-s + 7·49-s − 16·53-s + 16·55-s − 4·59-s + 5·61-s + 4·65-s − 11·67-s + 16·71-s + 2·73-s − 12·77-s + 5·79-s − 8·83-s + 32·85-s + 24·89-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.13·7-s − 1.20·11-s − 0.277·13-s − 1.94·17-s − 0.458·19-s − 0.834·23-s + 25-s + 0.718·31-s − 2.02·35-s − 2.95·37-s + 1.21·43-s + 1.75·47-s + 49-s − 2.19·53-s + 2.15·55-s − 0.520·59-s + 0.640·61-s + 0.496·65-s − 1.34·67-s + 1.89·71-s + 0.234·73-s − 1.36·77-s + 0.562·79-s − 0.878·83-s + 3.47·85-s + 2.54·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5305916845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5305916845\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 8 T - 19 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−11.12573372628180625021553837156, −10.34279613928047293802138897246, −10.28775364391878460688946722977, −9.129967806342950281007239561560, −9.089687989461646233211697428294, −8.356232687075987963188184794278, −8.197164984680305461817041930234, −7.68652719339444241270024505058, −7.52312942571354753703793902190, −6.85293717479024552172458505524, −6.48889313409536189724703290061, −5.67889180291999442741069502345, −5.16671411608697074323466965446, −4.62645817501812300502992141385, −4.33223844608278534144829883292, −3.85764727549165571248747817061, −3.17269579286039747651131008842, −2.32890127842941733712665714009, −1.88913075647490124066272569258, −0.37948990462083661397025196401,
0.37948990462083661397025196401, 1.88913075647490124066272569258, 2.32890127842941733712665714009, 3.17269579286039747651131008842, 3.85764727549165571248747817061, 4.33223844608278534144829883292, 4.62645817501812300502992141385, 5.16671411608697074323466965446, 5.67889180291999442741069502345, 6.48889313409536189724703290061, 6.85293717479024552172458505524, 7.52312942571354753703793902190, 7.68652719339444241270024505058, 8.197164984680305461817041930234, 8.356232687075987963188184794278, 9.089687989461646233211697428294, 9.129967806342950281007239561560, 10.28775364391878460688946722977, 10.34279613928047293802138897246, 11.12573372628180625021553837156
