| L(s) = 1 | + 2·3-s + 4-s + 3·9-s + 2·12-s + 2·13-s − 3·16-s + 12·17-s + 4·27-s + 12·29-s + 3·36-s + 4·39-s + 8·43-s − 6·48-s + 2·49-s + 24·51-s + 2·52-s − 12·53-s − 4·61-s − 7·64-s + 12·68-s − 16·79-s + 5·81-s + 24·87-s + 12·101-s − 16·103-s − 24·107-s + 4·108-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 9-s + 0.577·12-s + 0.554·13-s − 3/4·16-s + 2.91·17-s + 0.769·27-s + 2.22·29-s + 1/2·36-s + 0.640·39-s + 1.21·43-s − 0.866·48-s + 2/7·49-s + 3.36·51-s + 0.277·52-s − 1.64·53-s − 0.512·61-s − 7/8·64-s + 1.45·68-s − 1.80·79-s + 5/9·81-s + 2.57·87-s + 1.19·101-s − 1.57·103-s − 2.32·107-s + 0.384·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.559553292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.559553292\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.00467916272955544818941607342, −9.950883509439329918928548855049, −9.316193324389101243668353090925, −9.012525604542732739739282197235, −8.395421513078788980515219485775, −8.234756452490219312964017805000, −7.63570519375726352553717664062, −7.50631960058184002653580088982, −6.93548174906554347930325070034, −6.38538022154967726230096877373, −6.05462746589581197484607584344, −5.47194297151417843974516966826, −4.93266163837314642554371749871, −4.35179957296806594844870933078, −3.88265266079115475206211285074, −3.10809022980903272446505129450, −3.04262210108616053212904394019, −2.40868098729543463584390781909, −1.49808899212528247292178915742, −1.05953510937097008657874850035,
1.05953510937097008657874850035, 1.49808899212528247292178915742, 2.40868098729543463584390781909, 3.04262210108616053212904394019, 3.10809022980903272446505129450, 3.88265266079115475206211285074, 4.35179957296806594844870933078, 4.93266163837314642554371749871, 5.47194297151417843974516966826, 6.05462746589581197484607584344, 6.38538022154967726230096877373, 6.93548174906554347930325070034, 7.50631960058184002653580088982, 7.63570519375726352553717664062, 8.234756452490219312964017805000, 8.395421513078788980515219485775, 9.012525604542732739739282197235, 9.316193324389101243668353090925, 9.950883509439329918928548855049, 10.00467916272955544818941607342
