| L(s) = 1 | + 4·2-s + 3·3-s + 4·4-s + 14·5-s + 12·6-s − 45·7-s − 16·8-s + 2·9-s + 56·10-s + 5·11-s + 12·12-s − 180·14-s + 42·15-s − 64·16-s + 130·17-s + 8·18-s + 3·19-s + 56·20-s − 135·21-s + 20·22-s − 33·23-s − 48·24-s − 269·25-s − 171·27-s − 180·28-s + 198·29-s + 168·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 1/2·4-s + 1.25·5-s + 0.816·6-s − 2.42·7-s − 0.707·8-s + 2/27·9-s + 1.77·10-s + 0.137·11-s + 0.288·12-s − 3.43·14-s + 0.722·15-s − 16-s + 1.85·17-s + 0.104·18-s + 0.0362·19-s + 0.626·20-s − 1.40·21-s + 0.193·22-s − 0.299·23-s − 0.408·24-s − 2.15·25-s − 1.21·27-s − 1.21·28-s + 1.26·29-s + 1.02·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3051845567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3051845567\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - p T + 7 T^{2} + 52 p T^{3} - 968 T^{4} + 52 p^{4} T^{5} + 7 p^{6} T^{6} - p^{10} T^{7} + p^{12} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 7 T + 208 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 45 T + 887 T^{2} + 20340 T^{3} + 482820 T^{4} + 20340 p^{3} T^{5} + 887 p^{6} T^{6} + 45 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 5 T + 15 T^{2} + 13260 T^{3} - 1804736 T^{4} + 13260 p^{3} T^{5} + 15 p^{6} T^{6} - 5 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 130 T + 6321 T^{2} - 97890 T^{3} + 4747972 T^{4} - 97890 p^{3} T^{5} + 6321 p^{6} T^{6} - 130 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3 T - 13657 T^{2} + 156 T^{3} + 139651944 T^{4} + 156 p^{3} T^{5} - 13657 p^{6} T^{6} - 3 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 33 T - 23029 T^{2} - 7128 T^{3} + 420392172 T^{4} - 7128 p^{3} T^{5} - 23029 p^{6} T^{6} + 33 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 198 T - 11563 T^{2} - 393822 T^{3} + 1026318612 T^{4} - 393822 p^{3} T^{5} - 11563 p^{6} T^{6} - 198 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 280 T + 47934 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 702 T + 269165 T^{2} + 85877766 T^{3} + 22486674900 T^{4} + 85877766 p^{3} T^{5} + 269165 p^{6} T^{6} + 702 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 242 T - 72219 T^{2} + 1708278 T^{3} + 8317799404 T^{4} + 1708278 p^{3} T^{5} - 72219 p^{6} T^{6} - 242 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 93 T - 118621 T^{2} + 2952192 T^{3} + 9188633808 T^{4} + 2952192 p^{3} T^{5} - 118621 p^{6} T^{6} - 93 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 224 T + 50062 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 535 T + 369256 T^{2} + 535 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 389 T - 294609 T^{2} + 13681908 T^{3} + 125594875600 T^{4} + 13681908 p^{3} T^{5} - 294609 p^{6} T^{6} + 389 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 654 T - 77623 T^{2} + 33600558 T^{3} + 106763725164 T^{4} + 33600558 p^{3} T^{5} - 77623 p^{6} T^{6} + 654 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 107 T - 377621 T^{2} - 22732792 T^{3} + 57473647144 T^{4} - 22732792 p^{3} T^{5} - 377621 p^{6} T^{6} + 107 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 569 T - 453417 T^{2} + 34911564 T^{3} + 360027736492 T^{4} + 34911564 p^{3} T^{5} - 453417 p^{6} T^{6} + 569 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 623 T + 11820 p T^{2} - 623 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 760 T + 1005486 T^{2} - 760 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 1764 T + 1851190 T^{2} - 1764 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 871 T - 788823 T^{2} + 119785146 T^{3} + 1362845153206 T^{4} + 119785146 p^{3} T^{5} - 788823 p^{6} T^{6} + 871 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 879 T - 1154671 T^{2} + 89628114 T^{3} + 2176390961022 T^{4} + 89628114 p^{3} T^{5} - 1154671 p^{6} T^{6} + 879 p^{9} T^{7} + p^{12} T^{8} \) |
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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
−7.917671881008320168722620243585, −7.64152286519531170867674569401, −7.54866616895277878277750652037, −6.95846301490312941953692962931, −6.64962546525378239411985496046, −6.50873849590743717405456883487, −6.44629408110649901874504304600, −5.94379025214167271279365384433, −5.81633801972284012353740012102, −5.45720958099817849352065001702, −5.45237054727967764258885953150, −5.28511266468266724697675899643, −4.59252498981779404775298476950, −4.48842527074818064723364852967, −3.85342836382440461986473430692, −3.66732477674590375006431582290, −3.43622758298676762927042929691, −3.27713157306936143394590979460, −3.23062431828003314146417803046, −2.61287544835131675000899446151, −2.01663497500217691048363983357, −1.81684475025820271169101308142, −1.70623911850796226979215273031, −0.67462386446908347193800810222, −0.07434961302924994174188643875,
0.07434961302924994174188643875, 0.67462386446908347193800810222, 1.70623911850796226979215273031, 1.81684475025820271169101308142, 2.01663497500217691048363983357, 2.61287544835131675000899446151, 3.23062431828003314146417803046, 3.27713157306936143394590979460, 3.43622758298676762927042929691, 3.66732477674590375006431582290, 3.85342836382440461986473430692, 4.48842527074818064723364852967, 4.59252498981779404775298476950, 5.28511266468266724697675899643, 5.45237054727967764258885953150, 5.45720958099817849352065001702, 5.81633801972284012353740012102, 5.94379025214167271279365384433, 6.44629408110649901874504304600, 6.50873849590743717405456883487, 6.64962546525378239411985496046, 6.95846301490312941953692962931, 7.54866616895277878277750652037, 7.64152286519531170867674569401, 7.917671881008320168722620243585
