| L(s) = 1 | + 2-s − 3-s + 3·5-s − 6-s − 6·7-s + 8-s + 3·10-s − 11-s − 5·13-s − 6·14-s − 3·15-s − 16-s + 7·17-s − 19-s + 6·21-s − 22-s + 6·23-s − 24-s + 5·25-s − 5·26-s + 4·27-s + 29-s − 3·30-s − 5·31-s − 6·32-s + 33-s + 7·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1.34·5-s − 0.408·6-s − 2.26·7-s + 0.353·8-s + 0.948·10-s − 0.301·11-s − 1.38·13-s − 1.60·14-s − 0.774·15-s − 1/4·16-s + 1.69·17-s − 0.229·19-s + 1.30·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 25-s − 0.980·26-s + 0.769·27-s + 0.185·29-s − 0.547·30-s − 0.898·31-s − 1.06·32-s + 0.174·33-s + 1.20·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2697 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2697 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8051268543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8051268543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 + T + 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T - 2 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 9 T + 76 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−18.4396557267, −17.7657073855, −17.1832011489, −16.7015319237, −16.4141444387, −15.9967823643, −15.0081648140, −14.5840235394, −13.9255461708, −13.3913964776, −12.9859110694, −12.4783251424, −12.2131602091, −11.0817515723, −10.2982968232, −9.96080663030, −9.51509396807, −8.86436799199, −7.46469902723, −6.81190779592, −6.29212629851, −5.35684018825, −5.09639527132, −3.56905937275, −2.64617257729,
2.64617257729, 3.56905937275, 5.09639527132, 5.35684018825, 6.29212629851, 6.81190779592, 7.46469902723, 8.86436799199, 9.51509396807, 9.96080663030, 10.2982968232, 11.0817515723, 12.2131602091, 12.4783251424, 12.9859110694, 13.3913964776, 13.9255461708, 14.5840235394, 15.0081648140, 15.9967823643, 16.4141444387, 16.7015319237, 17.1832011489, 17.7657073855, 18.4396557267
