| L(s) = 1 | − 9-s + 4·11-s − 8·19-s − 18·29-s − 10·31-s + 2·41-s − 49-s − 10·59-s + 18·61-s − 8·71-s − 28·79-s + 81-s + 12·89-s − 4·99-s − 40·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 8·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.20·11-s − 1.83·19-s − 3.34·29-s − 1.79·31-s + 0.312·41-s − 1/7·49-s − 1.30·59-s + 2.30·61-s − 0.949·71-s − 3.15·79-s + 1/9·81-s + 1.27·89-s − 0.402·99-s − 3.83·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s + 0.611·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1805833537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1805833537\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 141 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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
−8.673439376450945520747205102697, −8.315335398409854385711632624173, −7.73755686042681509134389751216, −7.52588447013491603041007414627, −7.16720949506277819305322059337, −6.60216915659118008791952283135, −6.59293117927600950796400577837, −5.82073493205819988501089059668, −5.79668318685519340047573687303, −5.34063473122529919189993634473, −4.91312254290352212312082050450, −4.12274989867123386437061754686, −4.05618964430216416736997336134, −3.81181304629743126887099971694, −3.25096447993994385109726599201, −2.64268999502896984416793883310, −2.10641248325570228911291614073, −1.72225818579318422131643125827, −1.30887891703733151224605728958, −0.11625037179464037513676526104,
0.11625037179464037513676526104, 1.30887891703733151224605728958, 1.72225818579318422131643125827, 2.10641248325570228911291614073, 2.64268999502896984416793883310, 3.25096447993994385109726599201, 3.81181304629743126887099971694, 4.05618964430216416736997336134, 4.12274989867123386437061754686, 4.91312254290352212312082050450, 5.34063473122529919189993634473, 5.79668318685519340047573687303, 5.82073493205819988501089059668, 6.59293117927600950796400577837, 6.60216915659118008791952283135, 7.16720949506277819305322059337, 7.52588447013491603041007414627, 7.73755686042681509134389751216, 8.315335398409854385711632624173, 8.673439376450945520747205102697
