| L(s) = 1 | + 4-s + 4·11-s − 4·16-s − 20·29-s − 8·31-s − 4·41-s + 4·44-s + 20·59-s + 12·61-s − 5·64-s + 44·71-s + 32·79-s + 12·89-s + 48·101-s + 28·109-s − 20·116-s − 8·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.20·11-s − 16-s − 3.71·29-s − 1.43·31-s − 0.624·41-s + 0.603·44-s + 2.60·59-s + 1.53·61-s − 5/8·64-s + 5.22·71-s + 3.60·79-s + 1.27·89-s + 4.77·101-s + 2.68·109-s − 1.85·116-s − 0.727·121-s − 0.718·124-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 − 0.312·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.287805160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.287805160\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - T^{2} + 5 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $D_{4}$ | \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} - 114 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 p T^{2} + 659 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 10 T + 70 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7326 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 5030 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 178 T^{2} + 13331 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 36 T^{2} - 4010 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 22 T + 250 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 9150 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 16 T + 209 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 157 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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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
−7.44123834041897679903063706406, −7.41421904496491870950532291791, −7.21878542517889778180208383321, −6.80501896824354508123223281424, −6.55561364681611844453535830232, −6.39167326965225354527598354564, −6.35317451912469313849385528827, −5.96113950496678308390715207903, −5.50814986835270701986932434841, −5.47532730397400331580432098109, −5.05903544583601989574463104512, −5.05060326899302559607509480843, −4.80068737351865729137662196521, −4.08468834539344009947217866162, −3.98527890432724158685654259874, −3.84459755256625864013118812654, −3.52642671663629905397195125261, −3.42031498201743255397883523954, −3.03372936300460914596785233955, −2.16857170541265553963773382891, −2.10573064634070077603886748970, −2.04575594871964318394423956309, −1.86710467359163906338031227725, −0.76416499516960759110806516794, −0.67692972901026458455109863076,
0.67692972901026458455109863076, 0.76416499516960759110806516794, 1.86710467359163906338031227725, 2.04575594871964318394423956309, 2.10573064634070077603886748970, 2.16857170541265553963773382891, 3.03372936300460914596785233955, 3.42031498201743255397883523954, 3.52642671663629905397195125261, 3.84459755256625864013118812654, 3.98527890432724158685654259874, 4.08468834539344009947217866162, 4.80068737351865729137662196521, 5.05060326899302559607509480843, 5.05903544583601989574463104512, 5.47532730397400331580432098109, 5.50814986835270701986932434841, 5.96113950496678308390715207903, 6.35317451912469313849385528827, 6.39167326965225354527598354564, 6.55561364681611844453535830232, 6.80501896824354508123223281424, 7.21878542517889778180208383321, 7.41421904496491870950532291791, 7.44123834041897679903063706406
