L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s − 2·9-s + 3·11-s + 12-s + 16-s − 9·17-s + 2·18-s + 4·19-s − 3·22-s − 24-s − 25-s − 5·27-s − 32-s + 3·33-s + 9·34-s − 2·36-s − 4·38-s − 3·41-s − 2·43-s + 3·44-s + 48-s − 4·49-s + 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 1/4·16-s − 2.18·17-s + 0.471·18-s + 0.917·19-s − 0.639·22-s − 0.204·24-s − 1/5·25-s − 0.962·27-s − 0.176·32-s + 0.522·33-s + 1.54·34-s − 1/3·36-s − 0.648·38-s − 0.468·41-s − 0.304·43-s + 0.452·44-s + 0.144·48-s − 4/7·49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6447684142\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6447684142\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−12.65184638957528492562545189993, −11.76151110095666076243916527327, −11.48784336886980598015170905998, −10.96335506685343699596982946555, −10.11165591150437775767363928198, −9.458410160621127692185848356259, −8.916179542760820571893759379977, −8.556402588559235220073091676366, −7.79559470784852537439015929833, −6.94485930836828132546840783257, −6.44045207386154869417531163565, −5.48701651905825883013630977513, −4.34386851063787685085624782788, −3.28892661255911696961558158588, −2.12449507348446127913724547555,
2.12449507348446127913724547555, 3.28892661255911696961558158588, 4.34386851063787685085624782788, 5.48701651905825883013630977513, 6.44045207386154869417531163565, 6.94485930836828132546840783257, 7.79559470784852537439015929833, 8.556402588559235220073091676366, 8.916179542760820571893759379977, 9.458410160621127692185848356259, 10.11165591150437775767363928198, 10.96335506685343699596982946555, 11.48784336886980598015170905998, 11.76151110095666076243916527327, 12.65184638957528492562545189993