| L(s) = 1 | + 2-s − 2·3-s − 4-s − 2·6-s − 4·7-s − 3·8-s + 3·9-s + 8·11-s + 2·12-s − 13-s − 4·14-s − 16-s + 4·17-s + 3·18-s − 4·19-s + 8·21-s + 8·22-s − 8·23-s + 6·24-s − 6·25-s − 26-s − 4·27-s + 4·28-s − 4·29-s + 12·31-s + 5·32-s − 16·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.816·6-s − 1.51·7-s − 1.06·8-s + 9-s + 2.41·11-s + 0.577·12-s − 0.277·13-s − 1.06·14-s − 1/4·16-s + 0.970·17-s + 0.707·18-s − 0.917·19-s + 1.74·21-s + 1.70·22-s − 1.66·23-s + 1.22·24-s − 6/5·25-s − 0.196·26-s − 0.769·27-s + 0.755·28-s − 0.742·29-s + 2.15·31-s + 0.883·32-s − 2.78·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4456913213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4456913213\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−19.8934316645, −19.4604421777, −19.1551537949, −18.4282714936, −17.6707519500, −17.2849148009, −16.6272610009, −16.4135833599, −15.4732516475, −14.9380436528, −14.1819882616, −13.6745032861, −12.9568449998, −12.3172568607, −11.8538765065, −11.5820474614, −9.96467191573, −9.95280234664, −9.19274055467, −8.09899069409, −6.54556545043, −6.42897107073, −5.65486780280, −4.25303028693, −3.71205447024,
3.71205447024, 4.25303028693, 5.65486780280, 6.42897107073, 6.54556545043, 8.09899069409, 9.19274055467, 9.95280234664, 9.96467191573, 11.5820474614, 11.8538765065, 12.3172568607, 12.9568449998, 13.6745032861, 14.1819882616, 14.9380436528, 15.4732516475, 16.4135833599, 16.6272610009, 17.2849148009, 17.6707519500, 18.4282714936, 19.1551537949, 19.4604421777, 19.8934316645
