| L(s) = 1 | + 2·3-s + 4-s + 2·7-s + 9-s + 2·12-s + 16-s + 4·21-s + 25-s − 4·27-s + 2·28-s + 36-s + 4·37-s − 12·41-s + 4·43-s − 12·47-s + 2·48-s − 3·49-s + 12·59-s + 2·63-s + 64-s + 4·67-s + 2·75-s − 8·79-s − 11·81-s − 12·83-s + 4·84-s + 12·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.577·12-s + 1/4·16-s + 0.872·21-s + 1/5·25-s − 0.769·27-s + 0.377·28-s + 1/6·36-s + 0.657·37-s − 1.87·41-s + 0.609·43-s − 1.75·47-s + 0.288·48-s − 3/7·49-s + 1.56·59-s + 0.251·63-s + 1/8·64-s + 0.488·67-s + 0.230·75-s − 0.900·79-s − 1.22·81-s − 1.31·83-s + 0.436·84-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.272584661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272584661\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 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
−10.01266695836419241039470733145, −9.771090116951963747732462323505, −9.041719154707209699948703742671, −8.533257686385305165102164949969, −8.183724168811066492416186474591, −7.75597063544725054018966072028, −7.10808527443642834147576713133, −6.62466802890021473772322424591, −5.87067442401171255670989409057, −5.19911087118113613734413940025, −4.56791419165948345442304123775, −3.72205999427843870242915013881, −3.12199139294718950241930234600, −2.34514002285389325456479650283, −1.58323942624920489452226567445,
1.58323942624920489452226567445, 2.34514002285389325456479650283, 3.12199139294718950241930234600, 3.72205999427843870242915013881, 4.56791419165948345442304123775, 5.19911087118113613734413940025, 5.87067442401171255670989409057, 6.62466802890021473772322424591, 7.10808527443642834147576713133, 7.75597063544725054018966072028, 8.183724168811066492416186474591, 8.533257686385305165102164949969, 9.041719154707209699948703742671, 9.771090116951963747732462323505, 10.01266695836419241039470733145
