| L(s) = 1 | − 2-s − 4·3-s + 4-s − 3·5-s + 4·6-s − 4·7-s − 8-s + 7·9-s + 3·10-s − 2·11-s − 4·12-s + 4·14-s + 12·15-s + 16-s − 6·17-s − 7·18-s − 6·19-s − 3·20-s + 16·21-s + 2·22-s − 4·23-s + 4·24-s − 25-s − 4·27-s − 4·28-s − 12·30-s − 10·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.34·5-s + 1.63·6-s − 1.51·7-s − 0.353·8-s + 7/3·9-s + 0.948·10-s − 0.603·11-s − 1.15·12-s + 1.06·14-s + 3.09·15-s + 1/4·16-s − 1.45·17-s − 1.64·18-s − 1.37·19-s − 0.670·20-s + 3.49·21-s + 0.426·22-s − 0.834·23-s + 0.816·24-s − 1/5·25-s − 0.769·27-s − 0.755·28-s − 2.19·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 157 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 31 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + p T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 77 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 51 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T - 41 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−16.1925906223, −16.0442867886, −15.6603680254, −15.1140297945, −14.6534849715, −13.3746326883, −13.2846525045, −12.4757510971, −12.2005264823, −11.9014834355, −11.1775831333, −10.9072831040, −10.6623549922, −10.0000060428, −9.36610364541, −8.70837041678, −8.10736872113, −7.31121650737, −6.85947716005, −6.36139059159, −5.84128502965, −5.32463388803, −4.22679973040, −3.84155961266, −2.44812941182, 0, 0,
2.44812941182, 3.84155961266, 4.22679973040, 5.32463388803, 5.84128502965, 6.36139059159, 6.85947716005, 7.31121650737, 8.10736872113, 8.70837041678, 9.36610364541, 10.0000060428, 10.6623549922, 10.9072831040, 11.1775831333, 11.9014834355, 12.2005264823, 12.4757510971, 13.2846525045, 13.3746326883, 14.6534849715, 15.1140297945, 15.6603680254, 16.0442867886, 16.1925906223
