| L(s) = 1 | − 2-s − 4-s − 3·5-s + 7-s + 8-s − 2·9-s + 3·10-s + 3·11-s + 4·13-s − 14-s − 16-s + 2·18-s − 6·19-s + 3·20-s − 3·22-s + 8·25-s − 4·26-s + 3·27-s − 28-s − 2·29-s − 6·31-s + 5·32-s − 3·35-s + 2·36-s − 3·37-s + 6·38-s − 3·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.948·10-s + 0.904·11-s + 1.10·13-s − 0.267·14-s − 1/4·16-s + 0.471·18-s − 1.37·19-s + 0.670·20-s − 0.639·22-s + 8/5·25-s − 0.784·26-s + 0.577·27-s − 0.188·28-s − 0.371·29-s − 1.07·31-s + 0.883·32-s − 0.507·35-s + 1/3·36-s − 0.493·37-s + 0.973·38-s − 0.474·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2569247202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2569247202\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 148 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−19.7228696001, −19.2625330172, −18.8141373867, −18.1703438031, −17.7245770549, −16.8683974252, −16.6349961675, −15.7100777738, −15.1757603240, −14.4847639708, −13.9722591121, −13.0048256217, −12.3459379241, −11.4809073364, −11.1342787764, −10.3300672221, −8.98676232797, −8.78127121246, −8.19059310857, −7.14849199777, −6.11392088356, −4.60763755176, −3.67549431481,
3.67549431481, 4.60763755176, 6.11392088356, 7.14849199777, 8.19059310857, 8.78127121246, 8.98676232797, 10.3300672221, 11.1342787764, 11.4809073364, 12.3459379241, 13.0048256217, 13.9722591121, 14.4847639708, 15.1757603240, 15.7100777738, 16.6349961675, 16.8683974252, 17.7245770549, 18.1703438031, 18.8141373867, 19.2625330172, 19.7228696001
