L(s) = 1 | − 2·3-s + 3·4-s + 5-s + 9-s − 6·12-s − 2·15-s + 5·16-s + 3·20-s − 4·23-s − 4·25-s + 4·27-s − 4·31-s + 3·36-s + 45-s + 4·47-s − 10·48-s − 10·49-s + 18·53-s − 6·60-s + 3·64-s + 8·69-s + 8·75-s + 5·80-s − 11·81-s − 12·92-s + 8·93-s − 12·100-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 3/2·4-s + 0.447·5-s + 1/3·9-s − 1.73·12-s − 0.516·15-s + 5/4·16-s + 0.670·20-s − 0.834·23-s − 4/5·25-s + 0.769·27-s − 0.718·31-s + 1/2·36-s + 0.149·45-s + 0.583·47-s − 1.44·48-s − 1.42·49-s + 2.47·53-s − 0.774·60-s + 3/8·64-s + 0.963·69-s + 0.923·75-s + 0.559·80-s − 1.22·81-s − 1.25·92-s + 0.829·93-s − 6/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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
−7.24292596867903156031126681323, −6.85455537810511086070920096884, −6.31467956879432701214176906451, −6.12205185283256534755716218240, −5.75059269610149601501150262068, −5.42980993702063023403755773391, −4.93365822834711710094882709052, −4.37924684181327461533759245710, −3.76270187727379313723657177065, −3.35811893165568626215775057226, −2.56705481076741539590655747712, −2.28013523400158002708655585343, −1.69984557434057821377595788321, −1.04251462893649328175902331756, 0,
1.04251462893649328175902331756, 1.69984557434057821377595788321, 2.28013523400158002708655585343, 2.56705481076741539590655747712, 3.35811893165568626215775057226, 3.76270187727379313723657177065, 4.37924684181327461533759245710, 4.93365822834711710094882709052, 5.42980993702063023403755773391, 5.75059269610149601501150262068, 6.12205185283256534755716218240, 6.31467956879432701214176906451, 6.85455537810511086070920096884, 7.24292596867903156031126681323