| L(s) = 1 | + 2-s − 3·5-s − 7-s − 8-s − 3·10-s − 6·11-s + 13-s − 14-s − 16-s − 6·17-s + 4·19-s − 6·22-s − 6·23-s + 5·25-s + 26-s − 9·29-s + 10·31-s − 6·34-s + 3·35-s − 14·37-s + 4·38-s + 3·40-s + 6·41-s + 4·43-s − 6·46-s + 6·47-s + 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.34·5-s − 0.377·7-s − 0.353·8-s − 0.948·10-s − 1.80·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.27·22-s − 1.25·23-s + 25-s + 0.196·26-s − 1.67·29-s + 1.79·31-s − 1.02·34-s + 0.507·35-s − 2.30·37-s + 0.648·38-s + 0.474·40-s + 0.937·41-s + 0.609·43-s − 0.884·46-s + 0.875·47-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1874651509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1874651509\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.16521088711055462945433026023, −9.646340327924616780799044735719, −9.062859002339186922990670320013, −8.645155073176911816416612910334, −8.528183196034112520628861756681, −7.70518060214073985202439589059, −7.57574498324002078130841192484, −7.32559103687776293368506055623, −6.68691683329063027033795307091, −6.10538094323615451776469843420, −5.72923805517524127476159205205, −5.29973904621100719966743777549, −4.78143807462183318905549923389, −4.24975329947642706172953511492, −4.01754701711491797000059941657, −3.45912979118964922420805816391, −2.68808397404229597803729587407, −2.65469262308591094433482924622, −1.54331372373995417456064731116, −0.16476133390568287054959784593,
0.16476133390568287054959784593, 1.54331372373995417456064731116, 2.65469262308591094433482924622, 2.68808397404229597803729587407, 3.45912979118964922420805816391, 4.01754701711491797000059941657, 4.24975329947642706172953511492, 4.78143807462183318905549923389, 5.29973904621100719966743777549, 5.72923805517524127476159205205, 6.10538094323615451776469843420, 6.68691683329063027033795307091, 7.32559103687776293368506055623, 7.57574498324002078130841192484, 7.70518060214073985202439589059, 8.528183196034112520628861756681, 8.645155073176911816416612910334, 9.062859002339186922990670320013, 9.646340327924616780799044735719, 10.16521088711055462945433026023
