| L(s) = 1 | + 2·3-s + 2·5-s + 3·9-s + 4·11-s + 4·15-s + 8·23-s + 3·25-s + 4·27-s + 8·31-s + 8·33-s − 12·37-s + 6·45-s + 8·47-s + 2·49-s − 4·53-s + 8·55-s + 8·59-s + 8·67-s + 16·69-s + 16·71-s + 6·75-s + 5·81-s + 4·89-s + 16·93-s − 28·97-s + 12·99-s − 8·103-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 9-s + 1.20·11-s + 1.03·15-s + 1.66·23-s + 3/5·25-s + 0.769·27-s + 1.43·31-s + 1.39·33-s − 1.97·37-s + 0.894·45-s + 1.16·47-s + 2/7·49-s − 0.549·53-s + 1.07·55-s + 1.04·59-s + 0.977·67-s + 1.92·69-s + 1.89·71-s + 0.692·75-s + 5/9·81-s + 0.423·89-s + 1.65·93-s − 2.84·97-s + 1.20·99-s − 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.401542810\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.401542810\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.993496282302686206195198125716, −7.29201860743052216208452071264, −6.90288973309045662475015636022, −6.69737234413973131508820724953, −6.31965034308655948787214509814, −5.54633689805497384630359744363, −5.25727965094002063963903116451, −4.71603449530596003750730430315, −4.18262305461875357427739051957, −3.63794846247599968047002631294, −3.28812114749583791267647785141, −2.55622643130632353649065730858, −2.29155460910346097730788954315, −1.42030451954304777852594711436, −1.03548373679573166109153489584,
1.03548373679573166109153489584, 1.42030451954304777852594711436, 2.29155460910346097730788954315, 2.55622643130632353649065730858, 3.28812114749583791267647785141, 3.63794846247599968047002631294, 4.18262305461875357427739051957, 4.71603449530596003750730430315, 5.25727965094002063963903116451, 5.54633689805497384630359744363, 6.31965034308655948787214509814, 6.69737234413973131508820724953, 6.90288973309045662475015636022, 7.29201860743052216208452071264, 7.993496282302686206195198125716
