| L(s) = 1 | + 4·3-s − 4·5-s + 2·7-s + 11·9-s − 8·11-s + 12·13-s − 16·15-s + 8·19-s + 8·21-s − 4·23-s + 2·25-s + 20·27-s + 6·29-s − 32·33-s − 8·35-s − 10·37-s + 48·39-s + 20·41-s + 4·43-s − 44·45-s − 16·47-s + 3·49-s − 4·53-s + 32·55-s + 32·57-s + 8·59-s + 18·61-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 0.755·7-s + 11/3·9-s − 2.41·11-s + 3.32·13-s − 4.13·15-s + 1.83·19-s + 1.74·21-s − 0.834·23-s + 2/5·25-s + 3.84·27-s + 1.11·29-s − 5.57·33-s − 1.35·35-s − 1.64·37-s + 7.68·39-s + 3.12·41-s + 0.609·43-s − 6.55·45-s − 2.33·47-s + 3/7·49-s − 0.549·53-s + 4.31·55-s + 4.23·57-s + 1.04·59-s + 2.30·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.750221057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.750221057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 4 T^{3} - 20 T^{4} + 4 p T^{5} + 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 22 T^{3} - 68 T^{4} + 22 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 8 T + 41 T^{2} + 152 T^{3} + 532 T^{4} + 152 p T^{5} + 41 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 9 T^{2} - 84 T^{3} - 76 T^{4} - 84 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 41 T^{2} - 168 T^{3} + 644 T^{4} - 168 p T^{5} + 41 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 4 T + 53 T^{2} + 244 T^{3} + 1588 T^{4} + 244 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 1858 T^{4} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 10 T + 13 T^{2} + 130 T^{3} + 2500 T^{4} + 130 p T^{5} + 13 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 20 T + 125 T^{2} + 160 T^{3} - 5156 T^{4} + 160 p T^{5} + 125 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 4 T + 29 T^{2} + 372 T^{3} - 1492 T^{4} + 372 p T^{5} + 29 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 124 T^{3} + 1438 T^{4} + 124 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T + 65 T^{2} + 8 p T^{3} - 52 p T^{4} + 8 p^{2} T^{5} + 65 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 6 T - 47 T^{2} - 354 T^{3} - 204 T^{4} - 354 p T^{5} - 47 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 8 T + 17 T^{2} - 688 T^{3} - 8612 T^{4} - 688 p T^{5} + 17 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 68 T^{2} - 474 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 28614 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 8 T + 134 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 28 T + 197 T^{2} + 2408 T^{3} - 48788 T^{4} + 2408 p T^{5} + 197 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 6 T + 173 T^{2} + 966 T^{3} + 17676 T^{4} + 966 p T^{5} + 173 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.757723796705884927624622649650, −8.253779384272495290365209100204, −8.097190214778718078640450167220, −8.053137253504469657679272294914, −8.012884743024200980223629908603, −7.47796226374428617936103785903, −7.45762509760061540198987733183, −7.24702629035058128595045470691, −6.89046694179901144537616373774, −6.33799017358164502769520730446, −6.08927578737864437354592114204, −5.66388492060918873816781367191, −5.60760672594298418688906956480, −4.94898726315901955375340244502, −4.80972659012944149074876397327, −4.25400723953090574799473415763, −4.21770592090254312313077881580, −3.59446630296290096976854578788, −3.51646579956256495964805868985, −3.47450023298770843653119276078, −2.96304883255103036573203004353, −2.43032236749597662862668088796, −2.15928606466751431862994085829, −1.47360264029622721508204285114, −0.981538439482748637147898898564,
0.981538439482748637147898898564, 1.47360264029622721508204285114, 2.15928606466751431862994085829, 2.43032236749597662862668088796, 2.96304883255103036573203004353, 3.47450023298770843653119276078, 3.51646579956256495964805868985, 3.59446630296290096976854578788, 4.21770592090254312313077881580, 4.25400723953090574799473415763, 4.80972659012944149074876397327, 4.94898726315901955375340244502, 5.60760672594298418688906956480, 5.66388492060918873816781367191, 6.08927578737864437354592114204, 6.33799017358164502769520730446, 6.89046694179901144537616373774, 7.24702629035058128595045470691, 7.45762509760061540198987733183, 7.47796226374428617936103785903, 8.012884743024200980223629908603, 8.053137253504469657679272294914, 8.097190214778718078640450167220, 8.253779384272495290365209100204, 8.757723796705884927624622649650
