| L(s) = 1 | + 3·2-s − 4·3-s + 7·4-s + 5-s − 12·6-s − 7-s + 15·8-s + 10·9-s + 3·10-s + 11-s − 28·12-s − 6·13-s − 3·14-s − 4·15-s + 30·16-s + 2·17-s + 30·18-s − 12·19-s + 7·20-s + 4·21-s + 3·22-s − 17·23-s − 60·24-s + 5·25-s − 18·26-s − 20·27-s − 7·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 2.30·3-s + 7/2·4-s + 0.447·5-s − 4.89·6-s − 0.377·7-s + 5.30·8-s + 10/3·9-s + 0.948·10-s + 0.301·11-s − 8.08·12-s − 1.66·13-s − 0.801·14-s − 1.03·15-s + 15/2·16-s + 0.485·17-s + 7.07·18-s − 2.75·19-s + 1.56·20-s + 0.872·21-s + 0.639·22-s − 3.54·23-s − 12.2·24-s + 25-s − 3.53·26-s − 3.84·27-s − 1.32·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.389750587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.389750587\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 - 11 T + 111 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_4\times C_2$ | \( 1 - T - 4 T^{2} + 9 T^{3} + 11 T^{4} + 9 p T^{5} - 4 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - T - 10 T^{2} + 21 T^{3} + 89 T^{4} + 21 p T^{5} - 10 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_4\times C_2$ | \( 1 + 6 T + 23 T^{2} + 60 T^{3} + 61 T^{4} + 60 p T^{5} + 23 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 2 T + 7 T^{2} - 70 T^{3} + 441 T^{4} - 70 p T^{5} + 7 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 12 T + 35 T^{2} - 198 T^{3} - 1781 T^{4} - 198 p T^{5} + 35 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 17 T + 7 p T^{2} + 1121 T^{3} + 6044 T^{4} + 1121 p T^{5} + 7 p^{3} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 12 T + 65 T^{2} - 432 T^{3} + 3049 T^{4} - 432 p T^{5} + 65 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 20 T + 159 T^{2} - 670 T^{3} + 2591 T^{4} - 670 p T^{5} + 159 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 3 T + 42 T^{2} - 175 T^{3} + 891 T^{4} - 175 p T^{5} + 42 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_4\times C_2$ | \( 1 - 8 T + 21 T^{2} + 176 T^{3} - 2311 T^{4} + 176 p T^{5} + 21 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 7 T - 28 T^{2} + 185 T^{3} + 941 T^{4} + 185 p T^{5} - 28 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 19 T + 98 T^{2} - 105 T^{3} - 2869 T^{4} - 105 p T^{5} + 98 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 + 3 T - 50 T^{2} - 327 T^{3} + 1969 T^{4} - 327 p T^{5} - 50 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 11 T + 35 T^{2} + 589 T^{3} + 8344 T^{4} + 589 p T^{5} + 35 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 8 T - 33 T^{2} + 10 T^{3} + 4811 T^{4} + 10 p T^{5} - 33 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 13 T + 68 T^{2} + 981 T^{3} + 13325 T^{4} + 981 p T^{5} + 68 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + T + 135 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 17 T + 227 T^{2} + 17 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 6 T - 13 T^{2} + 768 T^{3} + 12565 T^{4} + 768 p T^{5} - 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 12 T + 117 T^{2} - 1780 T^{3} + 26181 T^{4} - 1780 p T^{5} + 117 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.22304327292002524603650375959, −6.71178733974087352362545171060, −6.56804044980286681773890516561, −6.37122635778031694597931044409, −6.25115268006496615724028735084, −6.06041630105963863905523489802, −6.05671063881638801380097856365, −5.89951017545658176256163659458, −5.54657954039778986196702707858, −4.88797239332999249229977114613, −4.81290853348086416796971788275, −4.77975197068403903747259275202, −4.71871293780915738423143863250, −4.21452907280946288649457105208, −4.15412835503068295023690357789, −4.02934327877206077403305503624, −3.54693070473579623272901077263, −3.01498013817282870733704656482, −2.67126195939673542077994256704, −2.50378683887455552660161232836, −2.33949420123929544863640609398, −1.92318652827187720568610952652, −1.30229468720426495030670909076, −1.21139248397315474640408340556, −0.44302273141209692034612821299,
0.44302273141209692034612821299, 1.21139248397315474640408340556, 1.30229468720426495030670909076, 1.92318652827187720568610952652, 2.33949420123929544863640609398, 2.50378683887455552660161232836, 2.67126195939673542077994256704, 3.01498013817282870733704656482, 3.54693070473579623272901077263, 4.02934327877206077403305503624, 4.15412835503068295023690357789, 4.21452907280946288649457105208, 4.71871293780915738423143863250, 4.77975197068403903747259275202, 4.81290853348086416796971788275, 4.88797239332999249229977114613, 5.54657954039778986196702707858, 5.89951017545658176256163659458, 6.05671063881638801380097856365, 6.06041630105963863905523489802, 6.25115268006496615724028735084, 6.37122635778031694597931044409, 6.56804044980286681773890516561, 6.71178733974087352362545171060, 7.22304327292002524603650375959
