| L(s) = 1 | − 4·2-s − 4·3-s + 8·4-s − 4·5-s + 16·6-s − 4·7-s − 8·8-s + 6·9-s + 16·10-s − 4·11-s − 32·12-s − 12·13-s + 16·14-s + 16·15-s − 4·16-s − 24·18-s − 4·19-s − 32·20-s + 16·21-s + 16·22-s + 32·24-s + 8·25-s + 48·26-s + 4·27-s − 32·28-s − 12·29-s − 64·30-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 4·4-s − 1.78·5-s + 6.53·6-s − 1.51·7-s − 2.82·8-s + 2·9-s + 5.05·10-s − 1.20·11-s − 9.23·12-s − 3.32·13-s + 4.27·14-s + 4.13·15-s − 16-s − 5.65·18-s − 0.917·19-s − 7.15·20-s + 3.49·21-s + 3.41·22-s + 6.53·24-s + 8/5·25-s + 9.41·26-s + 0.769·27-s − 6.04·28-s − 2.22·29-s − 11.6·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 324 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 68 T^{3} + 574 T^{4} + 68 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_4\times C_2$ | \( 1 + 28 T^{2} + 966 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 92 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 116 T^{3} + 1486 T^{4} + 116 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 660 T^{3} + 6046 T^{4} + 660 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 82 T^{2} + p^{2} T^{4} ) \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 236 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 14086 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} - 444 T^{3} - 12994 T^{4} - 444 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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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
−8.687419532715339279250843899999, −8.675153159905999938793793119879, −8.503019450685924762509530264929, −8.182201244163119704056554879272, −7.907216741793766775223809057454, −7.42806485205728782308563659016, −7.37146131685154598764561839780, −7.33355047843592947998284616037, −7.11917160694813217799537031794, −6.84336716455709595735069534256, −6.48391214539481755884914310108, −6.20035190925488296874818208898, −6.09512333626927729927754599689, −5.39539661036086140239566106526, −5.33002939648882355782904676472, −4.89285331386396891369699667375, −4.75449862668910935310008206388, −4.69124886545526011313388607680, −4.03931365284095312958045789388, −3.78105786482706271818575895923, −3.10795733396149695554613528780, −2.77563324287679678633895252305, −2.65273695317407281741397417057, −1.83390511900531902714314170877, −1.48238546821133835778694109709, 0, 0, 0, 0,
1.48238546821133835778694109709, 1.83390511900531902714314170877, 2.65273695317407281741397417057, 2.77563324287679678633895252305, 3.10795733396149695554613528780, 3.78105786482706271818575895923, 4.03931365284095312958045789388, 4.69124886545526011313388607680, 4.75449862668910935310008206388, 4.89285331386396891369699667375, 5.33002939648882355782904676472, 5.39539661036086140239566106526, 6.09512333626927729927754599689, 6.20035190925488296874818208898, 6.48391214539481755884914310108, 6.84336716455709595735069534256, 7.11917160694813217799537031794, 7.33355047843592947998284616037, 7.37146131685154598764561839780, 7.42806485205728782308563659016, 7.907216741793766775223809057454, 8.182201244163119704056554879272, 8.503019450685924762509530264929, 8.675153159905999938793793119879, 8.687419532715339279250843899999
