| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 3·7-s − 3·8-s − 9-s + 10-s − 4·11-s − 12-s − 2·13-s − 3·14-s + 15-s + 16-s + 2·17-s + 18-s − 20-s − 3·21-s + 4·22-s + 7·23-s + 3·24-s + 25-s + 2·26-s + 3·28-s + 2·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.13·7-s − 1.06·8-s − 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.223·20-s − 0.654·21-s + 0.852·22-s + 1.45·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.566·28-s + 0.371·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3119812484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3119812484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 22 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 7 T + 46 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 22 T + 226 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 26 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 86 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T - 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 112 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−19.6167315393, −19.0678416077, −18.4322208206, −18.0548382026, −17.5444219767, −16.9364116276, −16.5571996739, −15.5377798673, −15.2911002754, −14.7420648472, −13.8747836501, −13.1078940376, −12.1589642640, −11.7768464685, −11.2109757451, −10.5585702404, −9.87537115418, −8.80381500162, −8.28671043762, −7.53839704135, −6.67320054555, −5.48171420641, −4.81729047931, −2.84659053132,
2.84659053132, 4.81729047931, 5.48171420641, 6.67320054555, 7.53839704135, 8.28671043762, 8.80381500162, 9.87537115418, 10.5585702404, 11.2109757451, 11.7768464685, 12.1589642640, 13.1078940376, 13.8747836501, 14.7420648472, 15.2911002754, 15.5377798673, 16.5571996739, 16.9364116276, 17.5444219767, 18.0548382026, 18.4322208206, 19.0678416077, 19.6167315393
