| L(s) = 1 | − 2·2-s + 3·4-s + 4·7-s − 4·8-s − 9-s − 4·13-s − 8·14-s + 5·16-s + 2·18-s + 8·26-s + 12·28-s + 20·29-s − 6·32-s − 3·36-s − 16·37-s + 24·47-s − 2·49-s − 12·52-s − 16·56-s − 40·58-s + 4·61-s − 4·63-s + 7·64-s + 4·67-s + 4·72-s − 8·73-s + 32·74-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.51·7-s − 1.41·8-s − 1/3·9-s − 1.10·13-s − 2.13·14-s + 5/4·16-s + 0.471·18-s + 1.56·26-s + 2.26·28-s + 3.71·29-s − 1.06·32-s − 1/2·36-s − 2.63·37-s + 3.50·47-s − 2/7·49-s − 1.66·52-s − 2.13·56-s − 5.25·58-s + 0.512·61-s − 0.503·63-s + 7/8·64-s + 0.488·67-s + 0.471·72-s − 0.936·73-s + 3.71·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.388420604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.388420604\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.052545989215902396018189445234, −9.018004662881399509192646319293, −8.468212525277996482542995016108, −8.460347589898186358674868817946, −7.79743475197748041815268343052, −7.69540321604552579539324128677, −6.98716070297077277256115575438, −6.95939253208227105026924536217, −6.40540195270996919453928869184, −5.86414895295749421271660378898, −5.27490182008906782889769045452, −5.11119509728700263407936174600, −4.44317927517669356159017990914, −4.24293063716704901324541428581, −3.07503171863172281753892584264, −3.05789196260715498843694751148, −2.11052606940190957352736046954, −2.03842459472598429319864687694, −1.13070880181262797945631025726, −0.61376638006350763154283655686,
0.61376638006350763154283655686, 1.13070880181262797945631025726, 2.03842459472598429319864687694, 2.11052606940190957352736046954, 3.05789196260715498843694751148, 3.07503171863172281753892584264, 4.24293063716704901324541428581, 4.44317927517669356159017990914, 5.11119509728700263407936174600, 5.27490182008906782889769045452, 5.86414895295749421271660378898, 6.40540195270996919453928869184, 6.95939253208227105026924536217, 6.98716070297077277256115575438, 7.69540321604552579539324128677, 7.79743475197748041815268343052, 8.460347589898186358674868817946, 8.468212525277996482542995016108, 9.018004662881399509192646319293, 9.052545989215902396018189445234
