| L(s) = 1 | − 2·3-s + 2·5-s + 7-s − 3·9-s − 3·11-s + 2·13-s − 4·15-s − 18·17-s − 2·19-s − 2·21-s − 3·23-s + 25-s + 14·27-s + 3·29-s − 2·31-s + 6·33-s + 2·35-s − 16·37-s − 4·39-s + 6·41-s − 17·43-s − 6·45-s − 9·47-s + 6·49-s + 36·51-s − 6·55-s + 4·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.377·7-s − 9-s − 0.904·11-s + 0.554·13-s − 1.03·15-s − 4.36·17-s − 0.458·19-s − 0.436·21-s − 0.625·23-s + 1/5·25-s + 2.69·27-s + 0.557·29-s − 0.359·31-s + 1.04·33-s + 0.338·35-s − 2.63·37-s − 0.640·39-s + 0.937·41-s − 2.59·43-s − 0.894·45-s − 1.31·47-s + 6/7·49-s + 5.04·51-s − 0.809·55-s + 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1193242096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1193242096\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - T - 5 T^{2} + 8 T^{3} - 20 T^{4} + 8 p T^{5} - 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2 T + 10 T^{2} + 64 T^{3} - 185 T^{4} + 64 p T^{5} + 10 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + T + 30 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $D_4\times C_2$ | \( 1 - 3 T - 43 T^{2} + 18 T^{3} + 1602 T^{4} + 18 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 2 T - 26 T^{2} - 64 T^{3} - 185 T^{4} - 64 p T^{5} - 26 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 17 T + 139 T^{2} + 1088 T^{3} + 8224 T^{4} + 1088 p T^{5} + 139 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 9 T - 25 T^{2} + 108 T^{3} + 5220 T^{4} + 108 p T^{5} - 25 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3 T - 103 T^{2} + 18 T^{3} + 8532 T^{4} + 18 p T^{5} - 103 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 74 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 9 T - 97 T^{2} - 108 T^{3} + 18072 T^{4} - 108 p T^{5} - 97 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 15 T + 160 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 11 T - 95 T^{2} - 242 T^{3} + 25510 T^{4} - 242 p T^{5} - 95 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.23150275299571797326843847594, −7.07883326663211550189425493882, −7.07029882326073477280732772888, −6.56732739935333797512001845144, −6.42181100335918809527847325685, −6.31064795445806585845113464698, −6.28316653760480924353263066762, −5.87104830299926570490839906389, −5.55820279733234683070623559558, −5.25303668978348189925771308335, −5.24934320231587197705396059503, −4.90441204968841974734233093047, −4.56292427369574977162951961659, −4.42012588764006285357093255135, −4.41517139960761881157285576241, −3.71570529812278272961330773073, −3.33608604005334151096161475763, −3.25853268780041647815861755521, −2.86792122337565813396102077544, −2.19985562581886842298166770815, −2.16346493969925411353803671666, −2.06050521472812793312763179561, −1.67968990075918061043865538968, −0.74678151871686385370391301464, −0.12636429227923717942868969240,
0.12636429227923717942868969240, 0.74678151871686385370391301464, 1.67968990075918061043865538968, 2.06050521472812793312763179561, 2.16346493969925411353803671666, 2.19985562581886842298166770815, 2.86792122337565813396102077544, 3.25853268780041647815861755521, 3.33608604005334151096161475763, 3.71570529812278272961330773073, 4.41517139960761881157285576241, 4.42012588764006285357093255135, 4.56292427369574977162951961659, 4.90441204968841974734233093047, 5.24934320231587197705396059503, 5.25303668978348189925771308335, 5.55820279733234683070623559558, 5.87104830299926570490839906389, 6.28316653760480924353263066762, 6.31064795445806585845113464698, 6.42181100335918809527847325685, 6.56732739935333797512001845144, 7.07029882326073477280732772888, 7.07883326663211550189425493882, 7.23150275299571797326843847594
