| L(s) = 1 | − 3-s + 2·4-s − 5-s + 4·7-s − 6·11-s − 2·12-s + 4·13-s + 15-s − 7·19-s − 2·20-s − 4·21-s + 6·23-s + 27-s + 8·28-s − 3·29-s + 10·31-s + 6·33-s − 4·35-s + 16·37-s − 4·39-s + 6·41-s + 4·43-s − 12·44-s − 6·47-s − 2·49-s + 8·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s − 0.447·5-s + 1.51·7-s − 1.80·11-s − 0.577·12-s + 1.10·13-s + 0.258·15-s − 1.60·19-s − 0.447·20-s − 0.872·21-s + 1.25·23-s + 0.192·27-s + 1.51·28-s − 0.557·29-s + 1.79·31-s + 1.04·33-s − 0.676·35-s + 2.63·37-s − 0.640·39-s + 0.937·41-s + 0.609·43-s − 1.80·44-s − 0.875·47-s − 2/7·49-s + 1.10·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.558928477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.558928477\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 9 T + 10 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−11.70522320429231655425850609173, −11.34438472590641175599383608922, −11.28145519946896237305168221861, −10.84747121882814826328083154057, −10.51313632782297576579580533779, −9.946350406778203688306535106703, −9.137219930994872527620081041512, −8.434217602750645763792296431681, −8.165019330100892285752731708959, −7.79571012367595495667848250977, −7.29335768840567224342910229300, −6.50115687655823481505371873564, −6.21324663936253785246842856423, −5.57721125838198067926470722400, −4.78455255837517191829356593255, −4.65506023736418719204204806910, −3.73356401942851294073006599473, −2.56288290638956769447879286672, −2.34351914499770241363312561096, −1.03780072709899155447212920056,
1.03780072709899155447212920056, 2.34351914499770241363312561096, 2.56288290638956769447879286672, 3.73356401942851294073006599473, 4.65506023736418719204204806910, 4.78455255837517191829356593255, 5.57721125838198067926470722400, 6.21324663936253785246842856423, 6.50115687655823481505371873564, 7.29335768840567224342910229300, 7.79571012367595495667848250977, 8.165019330100892285752731708959, 8.434217602750645763792296431681, 9.137219930994872527620081041512, 9.946350406778203688306535106703, 10.51313632782297576579580533779, 10.84747121882814826328083154057, 11.28145519946896237305168221861, 11.34438472590641175599383608922, 11.70522320429231655425850609173
