| L(s) = 1 | + 2-s − 3-s − 4-s − 5-s − 6-s − 3·8-s − 2·9-s − 10-s + 3·11-s + 12-s + 15-s − 16-s + 8·17-s − 2·18-s − 8·19-s + 20-s + 3·22-s − 4·23-s + 3·24-s − 2·25-s + 2·27-s + 12·29-s + 30-s + 5·32-s − 3·33-s + 8·34-s + 2·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.06·8-s − 2/3·9-s − 0.316·10-s + 0.904·11-s + 0.288·12-s + 0.258·15-s − 1/4·16-s + 1.94·17-s − 0.471·18-s − 1.83·19-s + 0.223·20-s + 0.639·22-s − 0.834·23-s + 0.612·24-s − 2/5·25-s + 0.384·27-s + 2.22·29-s + 0.182·30-s + 0.883·32-s − 0.522·33-s + 1.37·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5545856602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5545856602\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−19.4824705242, −19.1551537949, −18.3971489242, −17.6707519500, −17.4769707622, −16.6272610009, −16.4384669514, −15.4732516475, −14.8783607055, −14.2224312943, −14.1819882616, −13.1968432040, −12.3172568607, −12.2305799850, −11.5820474614, −10.8219386531, −9.96467191573, −9.36526770843, −8.24864276848, −8.09899069409, −6.42897107073, −6.16821463973, −5.09766059610, −4.25303028693, −3.25423179226,
3.25423179226, 4.25303028693, 5.09766059610, 6.16821463973, 6.42897107073, 8.09899069409, 8.24864276848, 9.36526770843, 9.96467191573, 10.8219386531, 11.5820474614, 12.2305799850, 12.3172568607, 13.1968432040, 14.1819882616, 14.2224312943, 14.8783607055, 15.4732516475, 16.4384669514, 16.6272610009, 17.4769707622, 17.6707519500, 18.3971489242, 19.1551537949, 19.4824705242
