| L(s) = 1 | − 2·2-s + 3·4-s − 4·5-s + 4·7-s − 2·8-s + 8·10-s + 2·11-s + 8·13-s − 8·14-s − 6·17-s − 6·19-s − 12·20-s − 4·22-s − 14·23-s + 14·25-s − 16·26-s + 12·28-s + 4·29-s + 6·32-s + 12·34-s − 16·35-s + 2·37-s + 12·38-s + 8·40-s + 16·41-s − 28·43-s + 6·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.78·5-s + 1.51·7-s − 0.707·8-s + 2.52·10-s + 0.603·11-s + 2.21·13-s − 2.13·14-s − 1.45·17-s − 1.37·19-s − 2.68·20-s − 0.852·22-s − 2.91·23-s + 14/5·25-s − 3.13·26-s + 2.26·28-s + 0.742·29-s + 1.06·32-s + 2.05·34-s − 2.70·35-s + 0.328·37-s + 1.94·38-s + 1.26·40-s + 2.49·41-s − 4.26·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4605434191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4605434191\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + T^{2} - p T^{3} - 3 T^{4} - p^{2} T^{5} + p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T - 5 T^{2} + 42 T^{3} + 780 T^{4} + 42 p T^{5} - 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 7 T^{2} - 54 T^{3} - 204 T^{4} - 54 p T^{5} + 7 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 41 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 30 T^{2} - 61 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 2 T + T^{2} + 142 T^{3} - 1508 T^{4} + 142 p T^{5} + p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 14 T + 61 T^{2} - 574 T^{3} + 6804 T^{4} - 574 p T^{5} + 61 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 20 T + 202 T^{2} + 1840 T^{3} + 15195 T^{4} + 1840 p T^{5} + 202 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - p T^{2} - 138 T^{3} + 3420 T^{4} - 138 p T^{5} - p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T - 18 T^{2} - 336 T^{3} + 12107 T^{4} - 336 p T^{5} - 18 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 14 T + 183 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 12 T - 6 T^{2} + 48 T^{3} + 6659 T^{4} + 48 p T^{5} - 6 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 4 T - 18 T^{2} - 496 T^{3} - 6349 T^{4} - 496 p T^{5} - 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 22 T^{2} - 7437 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 6 T + 131 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.65527998088111704573187057164, −7.64685115753745201717314152599, −7.03965608739081474603488662510, −6.90348092888623271254759601293, −6.48737470772566669022260951376, −6.48378498101245259094655588796, −6.41509778388546869023153831698, −5.97743192435922022690245160041, −5.94868598386468812987435310381, −5.25994514068242824263688028027, −4.98677267985564683404868971079, −4.83662481099467662500769587868, −4.55574107111046491697669877589, −4.17610305447655481376708177679, −4.05984568664831566610473465616, −3.82704953159946730164569158429, −3.73115674347730912919928512399, −3.19269249959945522914473537213, −2.78405705825801798717083721270, −2.38695914952170782541225212293, −2.11060010848395593634293077390, −1.58668567852865526633602000635, −1.46885159013585808002176385066, −1.06995124577920140136894328316, −0.26349781849665998357254841468,
0.26349781849665998357254841468, 1.06995124577920140136894328316, 1.46885159013585808002176385066, 1.58668567852865526633602000635, 2.11060010848395593634293077390, 2.38695914952170782541225212293, 2.78405705825801798717083721270, 3.19269249959945522914473537213, 3.73115674347730912919928512399, 3.82704953159946730164569158429, 4.05984568664831566610473465616, 4.17610305447655481376708177679, 4.55574107111046491697669877589, 4.83662481099467662500769587868, 4.98677267985564683404868971079, 5.25994514068242824263688028027, 5.94868598386468812987435310381, 5.97743192435922022690245160041, 6.41509778388546869023153831698, 6.48378498101245259094655588796, 6.48737470772566669022260951376, 6.90348092888623271254759601293, 7.03965608739081474603488662510, 7.64685115753745201717314152599, 7.65527998088111704573187057164
