| L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s + 16·6-s − 4·7-s − 12·8-s + 12·9-s − 4·11-s − 32·12-s + 16·14-s + 15·16-s − 48·18-s + 8·19-s + 16·21-s + 16·22-s + 4·23-s + 48·24-s − 2·25-s − 20·27-s − 32·28-s − 4·29-s − 12·31-s − 16·32-s + 16·33-s + 96·36-s − 32·38-s − 4·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 4·4-s + 6.53·6-s − 1.51·7-s − 4.24·8-s + 4·9-s − 1.20·11-s − 9.23·12-s + 4.27·14-s + 15/4·16-s − 11.3·18-s + 1.83·19-s + 3.49·21-s + 3.41·22-s + 0.834·23-s + 9.79·24-s − 2/5·25-s − 3.84·27-s − 6.04·28-s − 0.742·29-s − 2.15·31-s − 2.82·32-s + 2.78·33-s + 16·36-s − 5.19·38-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83521 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02073318126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02073318126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + p^{3} T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{3} T^{5} + p^{5} T^{6} + p^{5} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 4 p T^{3} - 40 T^{4} - 4 p^{2} T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 2 T^{2} - 16 T^{3} + 2 T^{4} - 16 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 4 T + 4 T^{2} - 4 p T^{3} - 104 T^{4} - 4 p^{2} T^{5} + 4 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 12 T^{2} + 60 T^{3} + 184 T^{4} + 60 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 184 T^{3} + 1042 T^{4} - 184 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T + 12 T^{2} + 164 T^{3} - 712 T^{4} + 164 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T + 22 T^{2} + 244 T^{3} + 930 T^{4} + 244 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 12 T + 108 T^{2} + 804 T^{3} + 5112 T^{4} + 804 p T^{5} + 108 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 50 T^{2} - 240 T^{3} + 1250 T^{4} - 240 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 376 T^{3} + 4402 T^{4} + 376 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2854 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $D_4\times C_2$ | \( 1 + 50 T^{2} - 720 T^{3} + 1250 T^{4} - 720 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 20 T + 100 T^{2} + 20 p T^{3} - 23400 T^{4} + 20 p^{2} T^{5} + 100 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 28 T + 294 T^{2} + 1372 T^{3} + 4802 T^{4} + 1372 p T^{5} + 294 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T + 12 T^{2} - 836 T^{3} - 3400 T^{4} - 836 p T^{5} + 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 12466 T^{4} - 1264 p T^{5} + 128 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 27874 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 24 T + 242 T^{2} - 1704 T^{3} + 13730 T^{4} - 1704 p T^{5} + 242 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−15.27820992234551772161971224693, −14.29692826949536558292876311602, −13.87946730539711479922404677570, −13.08640749870537083344509537076, −13.01438382222935165276952421923, −12.64612980251574111590139919250, −12.41733962896031951736224781186, −11.69015461759250798597364696764, −11.57899485898614895591759569503, −11.10378052582036485546872080009, −10.74244975396732498786477290357, −10.31660880773855664947822962414, −10.06336598180932952144978764529, −9.734396513385092223719084388957, −9.538072377630503931469175384201, −8.833155096843014552661580438105, −8.787455779585150857889124623441, −7.59353250379502107111273587820, −7.58544203495244752743489520942, −7.04494793081621256455477703736, −6.59514495921055644029448722027, −6.14743572620168908564155996007, −5.27609823497030340629124026891, −5.15470058723758459368169597079, −3.48616011786062289576592162458,
3.48616011786062289576592162458, 5.15470058723758459368169597079, 5.27609823497030340629124026891, 6.14743572620168908564155996007, 6.59514495921055644029448722027, 7.04494793081621256455477703736, 7.58544203495244752743489520942, 7.59353250379502107111273587820, 8.787455779585150857889124623441, 8.833155096843014552661580438105, 9.538072377630503931469175384201, 9.734396513385092223719084388957, 10.06336598180932952144978764529, 10.31660880773855664947822962414, 10.74244975396732498786477290357, 11.10378052582036485546872080009, 11.57899485898614895591759569503, 11.69015461759250798597364696764, 12.41733962896031951736224781186, 12.64612980251574111590139919250, 13.01438382222935165276952421923, 13.08640749870537083344509537076, 13.87946730539711479922404677570, 14.29692826949536558292876311602, 15.27820992234551772161971224693
