| L(s) = 1 | + 4·7-s + 20·13-s + 28·19-s − 10·25-s − 8·31-s + 20·37-s − 152·43-s − 186·49-s − 100·61-s − 284·67-s + 164·73-s + 76·79-s + 80·91-s + 20·97-s + 100·103-s + 248·109-s + 124·121-s + 127-s + 131-s + 112·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 4/7·7-s + 1.53·13-s + 1.47·19-s − 2/5·25-s − 0.258·31-s + 0.540·37-s − 3.53·43-s − 3.79·49-s − 1.63·61-s − 4.23·67-s + 2.24·73-s + 0.962·79-s + 0.879·91-s + 0.206·97-s + 0.970·103-s + 2.27·109-s + 1.02·121-s + 0.00787·127-s + 0.00763·131-s + 0.842·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.753758070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.753758070\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 10 T + 3 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 506 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 14 T + 411 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1180 T^{2} + 700422 T^{4} - 1180 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2860 T^{2} + 3407622 T^{4} - 2860 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 486 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 10 T + 2403 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2764 T^{2} + 6265446 T^{4} - 2764 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 76 T + 3702 T^{2} + 76 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 3796 T^{2} + 8177766 T^{4} - 3796 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 788 T^{2} + 11737158 T^{4} + 788 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 6890 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 50 T + 2307 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 71 T + p^{2} T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4108 T^{2} - 8461722 T^{4} - 4108 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 82 T + 11979 T^{2} - 82 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 38 T + 3843 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2860 T^{2} - 27506298 T^{4} - 2860 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8500 T^{2} + 43184742 T^{4} - 8500 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 10 T + 9843 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.18495232679088504739467416329, −5.99156223746308432532786381251, −5.83862117625269490456209145751, −5.82222448265045612275375802269, −5.47458284324530327712570423046, −5.07760150052092594302480328106, −4.86137262699263793778896126981, −4.81187294954066147512841215887, −4.58327656710794239703475110220, −4.56422648896260737249973640007, −4.16406156022060616963289090126, −3.63281774689853563112723486826, −3.51742181365757417959803786801, −3.49345088315036323867723304509, −3.19860559882417072574132527397, −2.96364132159082060686103663904, −2.94984265181886267038961321384, −2.23417507735417806680388901924, −2.00586954426064664645434456763, −1.70488372864534811606911064339, −1.64677964398243202243299864318, −1.25705081487685245605574828306, −1.10502704315814740436573323873, −0.42581816998833457435218443018, −0.34844598642026516687069602614,
0.34844598642026516687069602614, 0.42581816998833457435218443018, 1.10502704315814740436573323873, 1.25705081487685245605574828306, 1.64677964398243202243299864318, 1.70488372864534811606911064339, 2.00586954426064664645434456763, 2.23417507735417806680388901924, 2.94984265181886267038961321384, 2.96364132159082060686103663904, 3.19860559882417072574132527397, 3.49345088315036323867723304509, 3.51742181365757417959803786801, 3.63281774689853563112723486826, 4.16406156022060616963289090126, 4.56422648896260737249973640007, 4.58327656710794239703475110220, 4.81187294954066147512841215887, 4.86137262699263793778896126981, 5.07760150052092594302480328106, 5.47458284324530327712570423046, 5.82222448265045612275375802269, 5.83862117625269490456209145751, 5.99156223746308432532786381251, 6.18495232679088504739467416329
