| L(s) = 1 | − 2·2-s − 3-s + 4-s − 3·5-s + 2·6-s + 10·7-s + 2·8-s + 3·9-s + 6·10-s + 9·11-s − 12-s + 8·13-s − 20·14-s + 3·15-s − 4·16-s − 6·18-s − 12·19-s − 3·20-s − 10·21-s − 18·22-s − 3·23-s − 2·24-s + 5·25-s − 16·26-s − 8·27-s + 10·28-s − 6·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s + 3.77·7-s + 0.707·8-s + 9-s + 1.89·10-s + 2.71·11-s − 0.288·12-s + 2.21·13-s − 5.34·14-s + 0.774·15-s − 16-s − 1.41·18-s − 2.75·19-s − 0.670·20-s − 2.18·21-s − 3.83·22-s − 0.625·23-s − 0.408·24-s + 25-s − 3.13·26-s − 1.53·27-s + 1.88·28-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.004952068\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.004952068\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 9 T + 53 T^{2} - 234 T^{3} + 852 T^{4} - 234 p T^{5} + 53 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 3 T - 31 T^{2} - 18 T^{3} + 864 T^{4} - 18 p T^{5} - 31 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 396 p T^{5} + 71 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 6 T - 10 T^{2} + 132 T^{3} - 441 T^{4} + 132 p T^{5} - 10 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 9 T + 94 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 49 T^{2} + 660 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 18 T + 218 T^{2} + 1980 T^{3} + 14967 T^{4} + 1980 p T^{5} + 218 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3 T - 91 T^{2} + 18 T^{3} + 6714 T^{4} + 18 p T^{5} - 91 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 9 T + 17 T^{2} - 486 T^{3} - 4164 T^{4} - 486 p T^{5} + 17 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 4266 p T^{5} + 401 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 9 T + 143 T^{2} + 1044 T^{3} + 10776 T^{4} + 1044 p T^{5} + 143 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 19710 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 74 p T^{5} - 83 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 18051 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 3 T - 163 T^{2} + 18 T^{3} + 20862 T^{4} + 18 p T^{5} - 163 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 13 T + 162 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.101298713781177224477914644211, −8.714165136287436719251607465676, −8.321971761656126251589090004627, −8.281692311837272084729167176943, −8.177342146917017033140003054973, −7.946114334651656511953912994291, −7.43597084647507459916943322192, −7.40768360743818186095223884254, −7.28479412697082408594346692964, −6.34861727149625616978157756234, −6.27703381296364396863576202218, −6.20536078412565368655301007716, −6.15564028334702752848442541921, −5.19349205229589477537241010637, −5.01005213489748806786828141599, −4.49757116710914825427406617272, −4.32708014018596054336369281595, −4.20781049770100152978706835380, −4.02163189905127661480712420049, −3.83869120561194407699329693338, −2.88566775485323823750553858300, −1.95513667344711490004689839021, −1.69202001641479529898607694123, −1.30910617883881824048118268129, −1.08122459295320025299405115064,
1.08122459295320025299405115064, 1.30910617883881824048118268129, 1.69202001641479529898607694123, 1.95513667344711490004689839021, 2.88566775485323823750553858300, 3.83869120561194407699329693338, 4.02163189905127661480712420049, 4.20781049770100152978706835380, 4.32708014018596054336369281595, 4.49757116710914825427406617272, 5.01005213489748806786828141599, 5.19349205229589477537241010637, 6.15564028334702752848442541921, 6.20536078412565368655301007716, 6.27703381296364396863576202218, 6.34861727149625616978157756234, 7.28479412697082408594346692964, 7.40768360743818186095223884254, 7.43597084647507459916943322192, 7.946114334651656511953912994291, 8.177342146917017033140003054973, 8.281692311837272084729167176943, 8.321971761656126251589090004627, 8.714165136287436719251607465676, 9.101298713781177224477914644211
