L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 2·5-s + 4·6-s + 2·8-s + 2·9-s − 4·10-s − 2·11-s − 2·12-s + 4·13-s − 4·15-s − 4·16-s − 4·17-s − 4·18-s − 10·19-s + 2·20-s + 4·22-s − 8·23-s − 4·24-s + 6·25-s − 8·26-s + 8·27-s + 8·30-s + 4·31-s + 2·32-s + 4·33-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 1.63·6-s + 0.707·8-s + 2/3·9-s − 1.26·10-s − 0.603·11-s − 0.577·12-s + 1.10·13-s − 1.03·15-s − 16-s − 0.970·17-s − 0.942·18-s − 2.29·19-s + 0.447·20-s + 0.852·22-s − 1.66·23-s − 0.816·24-s + 6/5·25-s − 1.56·26-s + 1.53·27-s + 1.46·30-s + 0.718·31-s + 0.353·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1953650481\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1953650481\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
good | 3 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 8 T^{3} - 17 T^{4} - 8 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 8 p T^{5} - 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 64 T^{3} - 237 T^{4} - 64 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 10 T + 42 T^{2} + 200 T^{3} + 1103 T^{4} + 200 p T^{5} + 42 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 18 T^{2} + 304 T^{3} - 1957 T^{4} + 304 p T^{5} + 18 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 8 T - 38 T^{2} - 32 T^{3} + 4203 T^{4} - 32 p T^{5} - 38 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10 T - 38 T^{2} - 200 T^{3} + 9663 T^{4} - 200 p T^{5} - 38 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 6 T - 90 T^{2} + 24 T^{3} + 9959 T^{4} + 24 p T^{5} - 90 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T + 58 T^{2} + 704 T^{3} - 3797 T^{4} + 704 p T^{5} + 58 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4$ | \( ( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T - 18 T^{2} + 512 T^{3} - 3277 T^{4} + 512 p T^{5} - 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.93301970404005469022236361331, −6.71524132039146494495187675943, −6.66793935507070859952216502062, −6.44051708591140015663688149942, −6.21904175356128230193433121938, −6.20289312483282726747417484916, −5.83983371739778540394085798278, −5.53680126006469395160874514065, −5.08192896163729125521252635361, −5.07127180662542370636488856294, −4.96190310199200306036564025225, −4.61949779704998171692916017177, −4.26636334225929755746766076681, −4.21979447396846498572077054378, −3.87276448199071481435993295039, −3.32176166096398450224484290843, −3.30514515412560041918856082066, −2.91373245706495228419903590258, −2.27996233513956075957863307642, −2.21429530804320868089054660550, −1.98511233931925031107968086101, −1.47464238243376668959017661907, −1.32087215159018707272653887482, −0.68207113210597681583455588215, −0.19400707111230236578173742123,
0.19400707111230236578173742123, 0.68207113210597681583455588215, 1.32087215159018707272653887482, 1.47464238243376668959017661907, 1.98511233931925031107968086101, 2.21429530804320868089054660550, 2.27996233513956075957863307642, 2.91373245706495228419903590258, 3.30514515412560041918856082066, 3.32176166096398450224484290843, 3.87276448199071481435993295039, 4.21979447396846498572077054378, 4.26636334225929755746766076681, 4.61949779704998171692916017177, 4.96190310199200306036564025225, 5.07127180662542370636488856294, 5.08192896163729125521252635361, 5.53680126006469395160874514065, 5.83983371739778540394085798278, 6.20289312483282726747417484916, 6.21904175356128230193433121938, 6.44051708591140015663688149942, 6.66793935507070859952216502062, 6.71524132039146494495187675943, 6.93301970404005469022236361331