| L(s) = 1 | + 3-s + 5-s − 7-s + 3·11-s + 8·13-s + 15-s − 4·19-s − 21-s + 8·23-s + 5·25-s − 27-s − 6·29-s − 5·31-s + 3·33-s − 35-s − 8·37-s + 8·39-s + 16·41-s − 12·43-s + 10·47-s − 6·49-s − 9·53-s + 3·55-s − 4·57-s − 5·59-s + 10·61-s + 8·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 0.904·11-s + 2.21·13-s + 0.258·15-s − 0.917·19-s − 0.218·21-s + 1.66·23-s + 25-s − 0.192·27-s − 1.11·29-s − 0.898·31-s + 0.522·33-s − 0.169·35-s − 1.31·37-s + 1.28·39-s + 2.49·41-s − 1.82·43-s + 1.45·47-s − 6/7·49-s − 1.23·53-s + 0.404·55-s − 0.529·57-s − 0.650·59-s + 1.28·61-s + 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.280267482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.280267482\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
| 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
−11.68679580528501483916401391011, −11.04435583558840974334112606251, −10.87646019263452338569845838805, −10.70482832250080415851759366785, −9.683454707047541994380299294704, −9.416301749636659985187879958376, −8.913724464164031676128686744716, −8.654119078104937237731057740227, −8.275549315163550816236887973917, −7.46330659107816661056272453218, −6.75880415736497526996388806916, −6.70819660452786775820537604428, −5.74372940352540178256403532821, −5.73921741878505684235593132581, −4.66742853115751436336700555480, −4.04881298692674209535868395267, −3.43472352107102871488601814606, −3.04168786639307196021575513186, −1.92782916575196486636189341960, −1.20686075024325120967931599859,
1.20686075024325120967931599859, 1.92782916575196486636189341960, 3.04168786639307196021575513186, 3.43472352107102871488601814606, 4.04881298692674209535868395267, 4.66742853115751436336700555480, 5.73921741878505684235593132581, 5.74372940352540178256403532821, 6.70819660452786775820537604428, 6.75880415736497526996388806916, 7.46330659107816661056272453218, 8.275549315163550816236887973917, 8.654119078104937237731057740227, 8.913724464164031676128686744716, 9.416301749636659985187879958376, 9.683454707047541994380299294704, 10.70482832250080415851759366785, 10.87646019263452338569845838805, 11.04435583558840974334112606251, 11.68679580528501483916401391011
