| L(s) = 1 | − 4-s − 2·5-s + 3·7-s − 3·11-s − 3·13-s + 16-s − 4·17-s + 4·19-s + 2·20-s − 3·23-s − 2·25-s − 3·28-s + 29-s − 14·31-s − 6·35-s − 6·37-s − 14·41-s + 13·43-s + 3·44-s + 13·47-s + 5·49-s + 3·52-s + 53-s + 6·55-s + 8·59-s − 11·61-s − 64-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.894·5-s + 1.13·7-s − 0.904·11-s − 0.832·13-s + 1/4·16-s − 0.970·17-s + 0.917·19-s + 0.447·20-s − 0.625·23-s − 2/5·25-s − 0.566·28-s + 0.185·29-s − 2.51·31-s − 1.01·35-s − 0.986·37-s − 2.18·41-s + 1.98·43-s + 0.452·44-s + 1.89·47-s + 5/7·49-s + 0.416·52-s + 0.137·53-s + 0.809·55-s + 1.04·59-s − 1.40·61-s − 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52812 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52812 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 163 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 16 T + p T^{2} ) \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 14 T + 100 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 42 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 13 T + 117 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 13 T + 131 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 48 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 48 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 107 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T - 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 98 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 5 T + 148 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T + 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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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
−14.8478956543, −14.5131962568, −14.0064717310, −13.6313588782, −13.1840159758, −12.5260045202, −12.1494545054, −11.8147173277, −11.2245096910, −10.8235954306, −10.3983804965, −9.83829956067, −9.07146294125, −8.85697797378, −8.19241964562, −7.65406433794, −7.42715593604, −6.92389333506, −5.77562021610, −5.34219217862, −4.90385240998, −4.13754986399, −3.74375180065, −2.65708784124, −1.78672642474, 0,
1.78672642474, 2.65708784124, 3.74375180065, 4.13754986399, 4.90385240998, 5.34219217862, 5.77562021610, 6.92389333506, 7.42715593604, 7.65406433794, 8.19241964562, 8.85697797378, 9.07146294125, 9.83829956067, 10.3983804965, 10.8235954306, 11.2245096910, 11.8147173277, 12.1494545054, 12.5260045202, 13.1840159758, 13.6313588782, 14.0064717310, 14.5131962568, 14.8478956543
