| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s + 10·11-s + 16-s + 18-s + 2·19-s + 10·22-s + 6·25-s + 32-s + 36-s + 2·38-s − 16·41-s + 8·43-s + 10·44-s + 8·49-s + 6·50-s − 6·59-s + 64-s − 14·67-s + 72-s − 8·73-s + 2·76-s + 81-s − 16·82-s − 18·83-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s + 3.01·11-s + 1/4·16-s + 0.235·18-s + 0.458·19-s + 2.13·22-s + 6/5·25-s + 0.176·32-s + 1/6·36-s + 0.324·38-s − 2.49·41-s + 1.21·43-s + 1.50·44-s + 8/7·49-s + 0.848·50-s − 0.781·59-s + 1/8·64-s − 1.71·67-s + 0.117·72-s − 0.936·73-s + 0.229·76-s + 1/9·81-s − 1.76·82-s − 1.97·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.856544483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.856544483\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
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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
−8.783536737420058141512688781506, −8.535187192831245113091079394234, −7.54436263332675498703579276974, −7.29095748891929312921397174617, −6.77113477957161585896551263874, −6.40405974120795434814710755568, −6.07306039089763662113429224215, −5.37171998327179090192793208967, −4.76983340317191888619826831422, −4.27337503986595838227908027924, −3.83255306525026725527821070922, −3.37024208032222848650644201133, −2.65466838853958334199549406080, −1.54435447926757362991255376252, −1.24400642905426047708449322889,
1.24400642905426047708449322889, 1.54435447926757362991255376252, 2.65466838853958334199549406080, 3.37024208032222848650644201133, 3.83255306525026725527821070922, 4.27337503986595838227908027924, 4.76983340317191888619826831422, 5.37171998327179090192793208967, 6.07306039089763662113429224215, 6.40405974120795434814710755568, 6.77113477957161585896551263874, 7.29095748891929312921397174617, 7.54436263332675498703579276974, 8.535187192831245113091079394234, 8.783536737420058141512688781506
