| L(s) = 1 | − 4·4-s + 9-s + 4·16-s + 10·19-s + 24·29-s − 10·31-s − 4·36-s + 48·41-s − 11·49-s − 12·59-s − 4·61-s + 16·64-s + 48·71-s − 40·76-s − 26·79-s + 12·89-s − 14·109-s − 96·116-s + 22·121-s + 40·124-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1/3·9-s + 16-s + 2.29·19-s + 4.45·29-s − 1.79·31-s − 2/3·36-s + 7.49·41-s − 1.57·49-s − 1.56·59-s − 0.512·61-s + 2·64-s + 5.69·71-s − 4.58·76-s − 2.92·79-s + 1.27·89-s − 1.34·109-s − 8.91·116-s + 2·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.820815518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.820815518\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{2} - 285 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 25 T^{2} - 744 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 25 T^{2} - 4704 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84075259522825185955111820836, −7.81041586513481574412097072979, −7.58597406066573915460401618623, −7.09993691160973803001616038346, −6.81978965370785655899994689513, −6.75103492803242852751327424640, −6.32801687757455377943876200125, −6.05088864523935925593539434945, −5.92284392863499565909039897782, −5.46322416163517631314936764119, −5.27490307712484451120293809952, −5.13611756408748007217213322032, −4.68336931999107110531549574532, −4.50102697823097930090977823458, −4.37514452189771802736098331047, −4.15772028007840777377121972315, −3.85570900508396731422554963367, −3.36061847049378953327760695490, −3.11217103927366761642988700715, −2.83645989017408715754363012947, −2.31623614856788694332784806084, −2.24511596005071443987992856487, −1.07493006068174258266583642518, −1.06932122275175570233074872956, −0.64195063720172979189651612427,
0.64195063720172979189651612427, 1.06932122275175570233074872956, 1.07493006068174258266583642518, 2.24511596005071443987992856487, 2.31623614856788694332784806084, 2.83645989017408715754363012947, 3.11217103927366761642988700715, 3.36061847049378953327760695490, 3.85570900508396731422554963367, 4.15772028007840777377121972315, 4.37514452189771802736098331047, 4.50102697823097930090977823458, 4.68336931999107110531549574532, 5.13611756408748007217213322032, 5.27490307712484451120293809952, 5.46322416163517631314936764119, 5.92284392863499565909039897782, 6.05088864523935925593539434945, 6.32801687757455377943876200125, 6.75103492803242852751327424640, 6.81978965370785655899994689513, 7.09993691160973803001616038346, 7.58597406066573915460401618623, 7.81041586513481574412097072979, 7.84075259522825185955111820836
