| L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s − 5·9-s + 4·10-s − 4·16-s + 4·17-s + 10·18-s + 8·19-s − 4·20-s − 4·23-s − 3·25-s + 12·31-s + 8·32-s − 8·34-s − 10·36-s + 2·37-s − 16·38-s − 8·41-s − 16·43-s + 10·45-s + 8·46-s + 8·47-s − 10·49-s + 6·50-s − 12·53-s + 8·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s − 5/3·9-s + 1.26·10-s − 16-s + 0.970·17-s + 2.35·18-s + 1.83·19-s − 0.894·20-s − 0.834·23-s − 3/5·25-s + 2.15·31-s + 1.41·32-s − 1.37·34-s − 5/3·36-s + 0.328·37-s − 2.59·38-s − 1.24·41-s − 2.43·43-s + 1.49·45-s + 1.17·46-s + 1.16·47-s − 1.42·49-s + 0.848·50-s − 1.64·53-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 484 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2042529109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2042529109\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−19.3922718580, −19.1857249719, −18.2764512729, −17.9414335735, −17.0336103204, −16.9483853593, −15.9140726033, −15.7772041936, −14.7397148174, −13.9818243124, −13.5686390571, −12.1137747715, −11.5293266281, −11.4512586103, −10.0355090972, −9.81375356518, −8.60353961929, −8.13266580748, −7.60852684016, −6.36261389471, −5.11227132196, −3.27381985667,
3.27381985667, 5.11227132196, 6.36261389471, 7.60852684016, 8.13266580748, 8.60353961929, 9.81375356518, 10.0355090972, 11.4512586103, 11.5293266281, 12.1137747715, 13.5686390571, 13.9818243124, 14.7397148174, 15.7772041936, 15.9140726033, 16.9483853593, 17.0336103204, 17.9414335735, 18.2764512729, 19.1857249719, 19.3922718580
