| L(s) = 1 | − 4-s + 9-s + 6·11-s − 3·16-s − 3·19-s − 5·25-s − 6·29-s − 8·31-s − 36-s + 6·41-s − 6·44-s − 10·49-s + 16·61-s + 7·64-s + 3·76-s + 4·79-s + 81-s + 18·89-s + 6·99-s + 5·100-s − 12·101-s − 8·109-s + 6·116-s + 14·121-s + 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1/3·9-s + 1.80·11-s − 3/4·16-s − 0.688·19-s − 25-s − 1.11·29-s − 1.43·31-s − 1/6·36-s + 0.937·41-s − 0.904·44-s − 1.42·49-s + 2.04·61-s + 7/8·64-s + 0.344·76-s + 0.450·79-s + 1/9·81-s + 1.90·89-s + 0.603·99-s + 1/2·100-s − 1.19·101-s − 0.766·109-s + 0.557·116-s + 1.27·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7634326245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7634326245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
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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
−12.48908770778049687592213385243, −11.75487660381291263851998363292, −11.31737705821078096522165339428, −10.79314931689723642396910058633, −9.806034063547401516806502246048, −9.376769457581374448442182543489, −8.996067894440421337157486513751, −8.234090339970131320869646808922, −7.39621793288213919926258092189, −6.72417571779454494289610368776, −6.10175724015351563393394441677, −5.15677525201464370358146477202, −4.12894588992759516904096102322, −3.75302538413385504615009042317, −1.93269559877427365661391325370,
1.93269559877427365661391325370, 3.75302538413385504615009042317, 4.12894588992759516904096102322, 5.15677525201464370358146477202, 6.10175724015351563393394441677, 6.72417571779454494289610368776, 7.39621793288213919926258092189, 8.234090339970131320869646808922, 8.996067894440421337157486513751, 9.376769457581374448442182543489, 9.806034063547401516806502246048, 10.79314931689723642396910058633, 11.31737705821078096522165339428, 11.75487660381291263851998363292, 12.48908770778049687592213385243
