| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·5-s − 2·6-s − 4·7-s + 8-s − 2·9-s − 2·10-s + 6·11-s − 2·12-s + 8·13-s − 4·14-s + 4·15-s + 16-s + 17-s − 2·18-s − 4·19-s − 2·20-s + 8·21-s + 6·22-s − 2·24-s − 6·25-s + 8·26-s + 10·27-s − 4·28-s − 10·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s − 0.632·10-s + 1.80·11-s − 0.577·12-s + 2.21·13-s − 1.06·14-s + 1.03·15-s + 1/4·16-s + 0.242·17-s − 0.471·18-s − 0.917·19-s − 0.447·20-s + 1.74·21-s + 1.27·22-s − 0.408·24-s − 6/5·25-s + 1.56·26-s + 1.92·27-s − 0.755·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4911619994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4911619994\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) |
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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
−19.9292301410, −19.1887245783, −19.1494492613, −18.1183629461, −17.4469801976, −16.7385636699, −16.6317948443, −15.7488207479, −15.7312805996, −14.5765256398, −14.2197849424, −13.2768755254, −12.8595646329, −11.9583441737, −11.7666126827, −10.9076921437, −10.8489466872, −9.28607198826, −8.95538623117, −7.77199473906, −6.65330731261, −5.99911855172, −5.87146418849, −3.90229547123, −3.67478222653,
3.67478222653, 3.90229547123, 5.87146418849, 5.99911855172, 6.65330731261, 7.77199473906, 8.95538623117, 9.28607198826, 10.8489466872, 10.9076921437, 11.7666126827, 11.9583441737, 12.8595646329, 13.2768755254, 14.2197849424, 14.5765256398, 15.7312805996, 15.7488207479, 16.6317948443, 16.7385636699, 17.4469801976, 18.1183629461, 19.1494492613, 19.1887245783, 19.9292301410
