| L(s) = 1 | + 3·3-s + 2·4-s − 5·7-s + 6·9-s + 6·12-s + 5·13-s − 8·19-s − 15·21-s + 10·25-s + 9·27-s − 10·28-s + 14·31-s + 12·36-s + 12·37-s + 15·39-s − 13·43-s + 18·49-s + 10·52-s − 24·57-s − 15·61-s − 30·63-s − 8·64-s − 21·67-s − 34·73-s + 30·75-s − 16·76-s + 26·79-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 4-s − 1.88·7-s + 2·9-s + 1.73·12-s + 1.38·13-s − 1.83·19-s − 3.27·21-s + 2·25-s + 1.73·27-s − 1.88·28-s + 2.51·31-s + 2·36-s + 1.97·37-s + 2.40·39-s − 1.98·43-s + 18/7·49-s + 1.38·52-s − 3.17·57-s − 1.92·61-s − 3.77·63-s − 64-s − 2.56·67-s − 3.97·73-s + 3.46·75-s − 1.83·76-s + 2.92·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.892026252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.892026252\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
| 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
−12.20451991458521757531125864133, −11.81304630716341968080759031897, −11.00985004001999004616389462856, −10.58921493789206326309490353247, −10.19824282904115939800797122599, −9.833109972530714809126527562638, −8.940005621434105175667922144109, −8.937869644527916290778562486477, −8.467749300075130034196741846508, −7.79933088963558749217157555327, −7.26179450250040413496658364663, −6.57095022960282410074537069827, −6.30326179667536165592530001401, −6.17911972433700601838042301730, −4.58740299506504913870549686021, −4.23442401067337627477533254992, −3.31460287835113398316659963278, −2.89219752080197295347989655315, −2.58903831305641193565273178659, −1.42830943399969465503204095885,
1.42830943399969465503204095885, 2.58903831305641193565273178659, 2.89219752080197295347989655315, 3.31460287835113398316659963278, 4.23442401067337627477533254992, 4.58740299506504913870549686021, 6.17911972433700601838042301730, 6.30326179667536165592530001401, 6.57095022960282410074537069827, 7.26179450250040413496658364663, 7.79933088963558749217157555327, 8.467749300075130034196741846508, 8.937869644527916290778562486477, 8.940005621434105175667922144109, 9.833109972530714809126527562638, 10.19824282904115939800797122599, 10.58921493789206326309490353247, 11.00985004001999004616389462856, 11.81304630716341968080759031897, 12.20451991458521757531125864133
