| 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
| \(L(3)\) |
\(\approx\) |
\(4.226089246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.226089246\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 222744 T + 24807444768 T^{2} + 222744 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.11647659177743144681102270323, −7.03572852298661747315649828590, −6.89308853135492999051287779068, −6.27144245958915250207967558229, −6.13386907633995523275945324460, −5.95173035394873221047136600179, −5.88106675042003646633123797862, −5.23171970212981927847898426832, −5.13512322894843218361437540014, −4.94031536488181266881368390583, −4.83184735989203142951257601236, −4.02950755883971969070810556062, −4.01574755046896553706845434679, −3.92962145747094699677963962720, −3.83365146329520660962710907628, −3.00996886301228493219329410531, −2.73974375777899201807874220445, −2.70478604581944258434014447689, −2.43096987662337213606644671286, −1.73777948564684608005836440693, −1.42595367831118096692178430648, −1.30265248973812050649252420353, −1.21913291304392450686839101410, −0.47554708731422934641334297685, −0.26543309934103991273783152134,
0.26543309934103991273783152134, 0.47554708731422934641334297685, 1.21913291304392450686839101410, 1.30265248973812050649252420353, 1.42595367831118096692178430648, 1.73777948564684608005836440693, 2.43096987662337213606644671286, 2.70478604581944258434014447689, 2.73974375777899201807874220445, 3.00996886301228493219329410531, 3.83365146329520660962710907628, 3.92962145747094699677963962720, 4.01574755046896553706845434679, 4.02950755883971969070810556062, 4.83184735989203142951257601236, 4.94031536488181266881368390583, 5.13512322894843218361437540014, 5.23171970212981927847898426832, 5.88106675042003646633123797862, 5.95173035394873221047136600179, 6.13386907633995523275945324460, 6.27144245958915250207967558229, 6.89308853135492999051287779068, 7.03572852298661747315649828590, 7.11647659177743144681102270323
