| L(s) = 1 | + 3-s + 2·4-s − 2·9-s + 2·12-s + 5·13-s − 6·17-s − 3·23-s − 25-s − 5·27-s + 6·29-s − 4·36-s + 5·39-s − 11·43-s + 5·49-s − 6·51-s + 10·52-s − 6·53-s + 13·61-s − 8·64-s − 12·68-s − 3·69-s − 75-s + 7·79-s + 81-s + 6·87-s − 6·92-s − 2·100-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 4-s − 2/3·9-s + 0.577·12-s + 1.38·13-s − 1.45·17-s − 0.625·23-s − 1/5·25-s − 0.962·27-s + 1.11·29-s − 2/3·36-s + 0.800·39-s − 1.67·43-s + 5/7·49-s − 0.840·51-s + 1.38·52-s − 0.824·53-s + 1.66·61-s − 64-s − 1.45·68-s − 0.361·69-s − 0.115·75-s + 0.787·79-s + 1/9·81-s + 0.643·87-s − 0.625·92-s − 1/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.462791507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462791507\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 157 T^{2} + p^{2} T^{4} \) |
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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
−11.06937929645315365520178910784, −10.93594436375944591737145655664, −10.22169940461072532546528114430, −9.497585477585207081976371047729, −8.871279211368385913124115833839, −8.373990587725152631748713218362, −8.059360947468859413416339897163, −7.10090601050224016178138442551, −6.53789513740480786554803192237, −6.19102450579877077584468932372, −5.34699002199756816358806463821, −4.34760802050208508977310815179, −3.54424435234633271901561057637, −2.69293404486165492040435902402, −1.89344939171266350594104110479,
1.89344939171266350594104110479, 2.69293404486165492040435902402, 3.54424435234633271901561057637, 4.34760802050208508977310815179, 5.34699002199756816358806463821, 6.19102450579877077584468932372, 6.53789513740480786554803192237, 7.10090601050224016178138442551, 8.059360947468859413416339897163, 8.373990587725152631748713218362, 8.871279211368385913124115833839, 9.497585477585207081976371047729, 10.22169940461072532546528114430, 10.93594436375944591737145655664, 11.06937929645315365520178910784
