L(s) = 1 | − 2-s − 3·3-s + 4-s − 5-s + 3·6-s − 2·7-s − 3·8-s + 3·9-s + 10-s − 3·11-s − 3·12-s + 2·14-s + 3·15-s + 16-s + 6·17-s − 3·18-s − 6·19-s − 20-s + 6·21-s + 3·22-s − 3·23-s + 9·24-s + 3·25-s − 2·28-s − 10·29-s − 3·30-s + 31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s − 0.755·7-s − 1.06·8-s + 9-s + 0.316·10-s − 0.904·11-s − 0.866·12-s + 0.534·14-s + 0.774·15-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 1.37·19-s − 0.223·20-s + 1.30·21-s + 0.639·22-s − 0.625·23-s + 1.83·24-s + 3/5·25-s − 0.377·28-s − 1.85·29-s − 0.547·30-s + 0.179·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3179 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 70 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 118 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 + 11 T + 66 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 18 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 138 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T - 26 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 46 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - T - 76 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 19 T + 246 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
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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
−18.5442796922, −17.7894353050, −17.3369791113, −16.8509015539, −16.4971637941, −16.0567369636, −15.4844073041, −14.9687431862, −14.3521307146, −13.2945260774, −12.7553104472, −12.1869122997, −11.9172420498, −11.1187737809, −10.7809006626, −10.2542041053, −9.43858091797, −8.91196293150, −7.87617027012, −7.40971289994, −6.34288927105, −5.96799060347, −5.42058720580, −4.17111998613, −2.82739846570, 0,
2.82739846570, 4.17111998613, 5.42058720580, 5.96799060347, 6.34288927105, 7.40971289994, 7.87617027012, 8.91196293150, 9.43858091797, 10.2542041053, 10.7809006626, 11.1187737809, 11.9172420498, 12.1869122997, 12.7553104472, 13.2945260774, 14.3521307146, 14.9687431862, 15.4844073041, 16.0567369636, 16.4971637941, 16.8509015539, 17.3369791113, 17.7894353050, 18.5442796922