| L(s) = 1 | + 4-s + 9-s + 4·19-s + 2·31-s + 36-s − 36·41-s + 13·49-s − 12·59-s − 16·61-s − 64-s + 12·71-s + 4·76-s + 22·79-s − 30·89-s − 36·101-s + 4·109-s + 22·121-s + 2·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 36·164-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 0.917·19-s + 0.359·31-s + 1/6·36-s − 5.62·41-s + 13/7·49-s − 1.56·59-s − 2.04·61-s − 1/8·64-s + 1.42·71-s + 0.458·76-s + 2.47·79-s − 3.17·89-s − 3.58·101-s + 0.383·109-s + 2·121-s + 0.179·124-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 − 2.81·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.663887603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663887603\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 85 T^{2} + 5016 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 - 50 T^{2} - 2829 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 145 T^{2} + p^{2} T^{4} )^{2} \) |
| 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
−7.10565078724765226656042585540, −6.86379708904958193680851990568, −6.66353989099662570627134765323, −6.42685922226398800158548323598, −6.41638841684716686796228736187, −6.01241091453760495191790695370, −5.66307821626685989616473577994, −5.44584949354032406631092878543, −5.29216167819234952519569431919, −5.14540267101443590581945578673, −4.81305534544587656852827914885, −4.62514114628719464529885456154, −4.24891457771657156706364606690, −4.05117917690613556700332084726, −3.78545961133479854033229205882, −3.47337731662721249504516270820, −3.02852656325557699168525087812, −3.01595787007257189767430984190, −2.96087516541886852215690935778, −2.23697661371650768255272198697, −1.97421038950503791588238376305, −1.71906922528736840031405767898, −1.43127376069712794799117051870, −1.00435969372114764072646419576, −0.28679888545285685383411831134,
0.28679888545285685383411831134, 1.00435969372114764072646419576, 1.43127376069712794799117051870, 1.71906922528736840031405767898, 1.97421038950503791588238376305, 2.23697661371650768255272198697, 2.96087516541886852215690935778, 3.01595787007257189767430984190, 3.02852656325557699168525087812, 3.47337731662721249504516270820, 3.78545961133479854033229205882, 4.05117917690613556700332084726, 4.24891457771657156706364606690, 4.62514114628719464529885456154, 4.81305534544587656852827914885, 5.14540267101443590581945578673, 5.29216167819234952519569431919, 5.44584949354032406631092878543, 5.66307821626685989616473577994, 6.01241091453760495191790695370, 6.41638841684716686796228736187, 6.42685922226398800158548323598, 6.66353989099662570627134765323, 6.86379708904958193680851990568, 7.10565078724765226656042585540
