| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s − 2·9-s + 2·11-s + 2·12-s + 16-s + 8·17-s + 2·18-s + 5·19-s − 2·22-s − 2·24-s + 25-s − 10·27-s − 32-s + 4·33-s − 8·34-s − 2·36-s − 5·38-s + 14·41-s + 3·43-s + 2·44-s + 2·48-s + 10·49-s − 50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s − 2/3·9-s + 0.603·11-s + 0.577·12-s + 1/4·16-s + 1.94·17-s + 0.471·18-s + 1.14·19-s − 0.426·22-s − 0.408·24-s + 1/5·25-s − 1.92·27-s − 0.176·32-s + 0.696·33-s − 1.37·34-s − 1/3·36-s − 0.811·38-s + 2.18·41-s + 0.457·43-s + 0.301·44-s + 0.288·48-s + 10/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2569600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2569600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.781773706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781773706\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 14 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.70750347707540122958265491717, −7.41955879647042832226307860928, −7.10711451402307345277270708225, −6.37590201938091478040645891504, −5.88340246680883398541021294321, −5.63563270581734548759772163737, −5.30579484401792830236791866391, −4.47391824840640186505086197265, −3.93360834185904133398812554518, −3.44137672968672308380535016792, −3.03580980777482528241632322051, −2.69442192521042929298433147903, −2.11117772225304628916970063943, −1.28013969129174437251913879613, −0.74883093863774344006289096180,
0.74883093863774344006289096180, 1.28013969129174437251913879613, 2.11117772225304628916970063943, 2.69442192521042929298433147903, 3.03580980777482528241632322051, 3.44137672968672308380535016792, 3.93360834185904133398812554518, 4.47391824840640186505086197265, 5.30579484401792830236791866391, 5.63563270581734548759772163737, 5.88340246680883398541021294321, 6.37590201938091478040645891504, 7.10711451402307345277270708225, 7.41955879647042832226307860928, 7.70750347707540122958265491717
