| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 7-s − 8-s − 2·9-s − 10-s + 11-s − 12-s + 13-s − 14-s − 15-s + 3·16-s − 3·17-s − 2·18-s − 3·19-s + 20-s − 21-s + 22-s + 3·23-s − 24-s − 3·25-s + 26-s − 2·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 3/4·16-s − 0.727·17-s − 0.471·18-s − 0.688·19-s + 0.223·20-s − 0.218·21-s + 0.213·22-s + 0.625·23-s − 0.204·24-s − 3/5·25-s + 0.196·26-s − 0.384·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3274 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3274 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9473855752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9473855752\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 1637 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 9 T + p T^{2} ) \) |
| good | 3 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 37 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 46 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 5 T + 67 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 19 T + 187 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 86 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 130 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−18.0890579960, −17.6080406068, −16.9855280524, −16.5581978887, −15.9476628051, −15.1974514914, −14.8384494523, −14.4276417658, −13.8177656926, −13.3290004411, −12.8501291653, −12.4222291922, −11.4996083755, −11.1306642030, −10.3702439679, −9.50970335914, −8.93049891948, −8.53998450266, −7.70307070827, −7.00269187515, −5.95992422818, −5.40362168031, −4.17319302046, −3.83770946754, −2.62935160096,
2.62935160096, 3.83770946754, 4.17319302046, 5.40362168031, 5.95992422818, 7.00269187515, 7.70307070827, 8.53998450266, 8.93049891948, 9.50970335914, 10.3702439679, 11.1306642030, 11.4996083755, 12.4222291922, 12.8501291653, 13.3290004411, 13.8177656926, 14.4276417658, 14.8384494523, 15.1974514914, 15.9476628051, 16.5581978887, 16.9855280524, 17.6080406068, 18.0890579960
