| L(s) = 1 | − 4-s − 6·9-s + 16-s − 4·17-s + 8·23-s − 25-s − 4·29-s + 6·36-s + 14·49-s + 12·53-s − 4·61-s − 64-s + 4·68-s − 16·79-s + 27·81-s − 8·92-s + 100-s − 12·101-s + 8·103-s − 28·113-s + 4·116-s + 22·121-s + 127-s + 131-s + 137-s + 139-s − 6·144-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2·9-s + 1/4·16-s − 0.970·17-s + 1.66·23-s − 1/5·25-s − 0.742·29-s + 36-s + 2·49-s + 1.64·53-s − 0.512·61-s − 1/8·64-s + 0.485·68-s − 1.80·79-s + 3·81-s − 0.834·92-s + 1/10·100-s − 1.19·101-s + 0.788·103-s − 2.63·113-s + 0.371·116-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1/2·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9188612017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9188612017\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.373331813564113357150496991026, −9.042159761669024467561706742086, −8.703482871178415330042041555047, −8.568016321569136926677194088464, −8.148819456316991103048834877022, −7.53339936156790697841854095173, −7.10753543583231997149896565352, −6.90096619849437747347419971221, −6.12314524184372262900906244962, −5.91010166909286343180898209832, −5.35490159334973590972798860809, −5.29229599758772307646267521891, −4.45862976027016903999514322779, −4.28430768095959156448137062313, −3.41240202173963204416407913261, −3.27001569363325189900705188871, −2.40089271930384107364385377342, −2.36997105049329329620760111974, −1.20290322255958635363240845710, −0.39526203760243125492212269677,
0.39526203760243125492212269677, 1.20290322255958635363240845710, 2.36997105049329329620760111974, 2.40089271930384107364385377342, 3.27001569363325189900705188871, 3.41240202173963204416407913261, 4.28430768095959156448137062313, 4.45862976027016903999514322779, 5.29229599758772307646267521891, 5.35490159334973590972798860809, 5.91010166909286343180898209832, 6.12314524184372262900906244962, 6.90096619849437747347419971221, 7.10753543583231997149896565352, 7.53339936156790697841854095173, 8.148819456316991103048834877022, 8.568016321569136926677194088464, 8.703482871178415330042041555047, 9.042159761669024467561706742086, 9.373331813564113357150496991026
