| L(s) = 1 | − 2-s − 4-s + 3·8-s − 9-s + 4·13-s − 16-s − 6·17-s + 18-s − 8·19-s + 6·25-s − 4·26-s − 5·32-s + 6·34-s + 36-s + 8·38-s − 4·43-s + 4·47-s − 2·49-s − 6·50-s − 4·52-s + 20·53-s − 4·59-s + 7·64-s − 24·67-s + 6·68-s − 3·72-s + 8·76-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1/3·9-s + 1.10·13-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.83·19-s + 6/5·25-s − 0.784·26-s − 0.883·32-s + 1.02·34-s + 1/6·36-s + 1.29·38-s − 0.609·43-s + 0.583·47-s − 2/7·49-s − 0.848·50-s − 0.554·52-s + 2.74·53-s − 0.520·59-s + 7/8·64-s − 2.93·67-s + 0.727·68-s − 0.353·72-s + 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
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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
−8.819525134841225778893794260369, −8.587152307806267041563523991254, −8.423252631377255836205143348714, −7.57500720647247100198180879119, −7.09764694772594784618536825224, −6.56781727993465695939560944623, −6.11602837457454352833300471898, −5.50930613378260735004203158068, −4.79102397113947667379949385953, −4.24174209127698662726129775577, −3.96650987522125204214984029616, −2.95702517646552303769679338272, −2.18776316937710797826324149823, −1.29392513804098680666116458679, 0,
1.29392513804098680666116458679, 2.18776316937710797826324149823, 2.95702517646552303769679338272, 3.96650987522125204214984029616, 4.24174209127698662726129775577, 4.79102397113947667379949385953, 5.50930613378260735004203158068, 6.11602837457454352833300471898, 6.56781727993465695939560944623, 7.09764694772594784618536825224, 7.57500720647247100198180879119, 8.423252631377255836205143348714, 8.587152307806267041563523991254, 8.819525134841225778893794260369
