| L(s) = 1 | + 2-s − 4-s + 5-s − 2·7-s − 3·8-s + 2·9-s + 10-s + 4·11-s − 6·13-s − 2·14-s − 16-s + 4·17-s + 2·18-s − 8·19-s − 20-s + 4·22-s − 4·23-s − 2·25-s − 6·26-s + 2·28-s + 2·29-s − 4·31-s + 5·32-s + 4·34-s − 2·35-s − 2·36-s − 8·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s − 1.06·8-s + 2/3·9-s + 0.316·10-s + 1.20·11-s − 1.66·13-s − 0.534·14-s − 1/4·16-s + 0.970·17-s + 0.471·18-s − 1.83·19-s − 0.223·20-s + 0.852·22-s − 0.834·23-s − 2/5·25-s − 1.17·26-s + 0.377·28-s + 0.371·29-s − 0.718·31-s + 0.883·32-s + 0.685·34-s − 0.338·35-s − 1/3·36-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3280 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3280 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8958033052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8958033052\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 10 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 6 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 62 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−18.1256721050, −17.4196855970, −17.0747116293, −16.7195336646, −15.9556272102, −15.3020051063, −14.6603725862, −14.5070517647, −13.8606204090, −13.3063291969, −12.5139430650, −12.4130745138, −11.9919676509, −10.8880259260, −10.1208824134, −9.65733976669, −9.28522108061, −8.49094104480, −7.53374937238, −6.76572487473, −6.09671014934, −5.41417855631, −4.30479547895, −3.89361585968, −2.45814248636,
2.45814248636, 3.89361585968, 4.30479547895, 5.41417855631, 6.09671014934, 6.76572487473, 7.53374937238, 8.49094104480, 9.28522108061, 9.65733976669, 10.1208824134, 10.8880259260, 11.9919676509, 12.4130745138, 12.5139430650, 13.3063291969, 13.8606204090, 14.5070517647, 14.6603725862, 15.3020051063, 15.9556272102, 16.7195336646, 17.0747116293, 17.4196855970, 18.1256721050
