| L(s) = 1 | − 132·7-s − 852·13-s − 256·16-s + 1.10e4·31-s + 1.78e4·37-s − 1.29e4·43-s + 8.71e3·49-s + 1.19e5·61-s + 7.94e4·67-s + 1.80e5·73-s + 1.12e5·91-s + 4.45e5·97-s + 4.39e5·103-s + 3.37e4·112-s + 5.10e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.62e5·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.01·7-s − 1.39·13-s − 1/4·16-s + 2.06·31-s + 2.14·37-s − 1.06·43-s + 0.518·49-s + 4.09·61-s + 2.16·67-s + 3.95·73-s + 1.42·91-s + 4.80·97-s + 4.08·103-s + 0.254·112-s + 3.16·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.977·169-s + 2.54e−6·173-s + 2.33e−6·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.175313230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.175313230\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 + p^{8} T^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 66 T + 2178 T^{2} + 66 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 255124 T^{2} + p^{10} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 426 T + 90738 T^{2} + 426 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 1785746551106 T^{4} + p^{20} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 p^{4} T^{2} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 50766012369602 T^{4} + p^{20} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 34585426 T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2764 T + p^{5} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 8934 T + 39908178 T^{2} - 8934 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 173457184 T^{2} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6480 T + 20995200 T^{2} + 6480 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 36780229269184702 T^{4} + p^{20} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 154847626826742002 T^{4} + p^{20} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 1298725780 T^{2} + p^{10} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 29750 T + p^{5} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 39732 T + 789315912 T^{2} - 39732 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 1473070970 T^{2} + p^{10} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 90132 T + 4061888712 T^{2} - 90132 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 6145160734 T^{2} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 7926948508975481426 T^{4} + p^{20} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 1392640 p^{2} T^{2} + p^{10} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 222744 T + 24807444768 T^{2} - 222744 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 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
−7.32157490647785955994309367755, −6.69059331396441445409866420086, −6.63226598078905504642035692826, −6.47935137760749944345765389971, −6.46600403511119770298395214960, −5.95736876497616870254661011374, −5.68047872153498884533336241590, −5.32619469500539990545816938805, −5.27392226745350409380051079779, −4.76064984296710647143375558451, −4.68264550155688099856942723977, −4.38026279340108110152564346455, −4.20948872548610671824892465948, −3.55844682421249064978045324893, −3.48668289338585945104165831048, −3.29052725047650176918374624037, −2.94243193411776356378855516456, −2.48282155599254261410947378466, −2.14456943494708693225050805248, −2.08076860388148764246787728469, −1.96035553739673211313585737535, −0.863140013044338302629209842820, −0.68839361787154385243698380845, −0.68276315387006668740000675667, −0.51149195291790588830123614423,
0.51149195291790588830123614423, 0.68276315387006668740000675667, 0.68839361787154385243698380845, 0.863140013044338302629209842820, 1.96035553739673211313585737535, 2.08076860388148764246787728469, 2.14456943494708693225050805248, 2.48282155599254261410947378466, 2.94243193411776356378855516456, 3.29052725047650176918374624037, 3.48668289338585945104165831048, 3.55844682421249064978045324893, 4.20948872548610671824892465948, 4.38026279340108110152564346455, 4.68264550155688099856942723977, 4.76064984296710647143375558451, 5.27392226745350409380051079779, 5.32619469500539990545816938805, 5.68047872153498884533336241590, 5.95736876497616870254661011374, 6.46600403511119770298395214960, 6.47935137760749944345765389971, 6.63226598078905504642035692826, 6.69059331396441445409866420086, 7.32157490647785955994309367755
