| L(s) = 1 | + 2·3-s + 6·5-s + 14·7-s + 2·9-s − 20·11-s + 32·13-s + 12·15-s + 16·17-s + 28·21-s + 38·23-s + 18·25-s − 22·27-s − 36·31-s − 40·33-s + 84·35-s + 60·37-s + 64·39-s + 52·41-s − 94·43-s + 12·45-s − 106·47-s + 98·49-s + 32·51-s + 12·53-s − 120·55-s + 204·61-s + 28·63-s + ⋯ |
| L(s) = 1 | + 2/3·3-s + 6/5·5-s + 2·7-s + 2/9·9-s − 1.81·11-s + 2.46·13-s + 4/5·15-s + 0.941·17-s + 4/3·21-s + 1.65·23-s + 0.719·25-s − 0.814·27-s − 1.16·31-s − 1.21·33-s + 12/5·35-s + 1.62·37-s + 1.64·39-s + 1.26·41-s − 2.18·43-s + 4/15·45-s − 2.25·47-s + 2·49-s + 0.627·51-s + 0.226·53-s − 2.18·55-s + 3.34·61-s + 4/9·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(11.06888503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.06888503\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} + 22 T^{3} - 158 T^{4} + 22 p^{2} T^{5} + 2 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 p T + 2 p^{2} T^{2} - 106 p T^{3} + 5602 T^{4} - 106 p^{3} T^{5} + 2 p^{6} T^{6} - 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 10 T + 226 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 6880 T^{3} + 90334 T^{4} - 6880 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 16 T + 128 T^{2} - 3824 T^{3} + 111742 T^{4} - 3824 p^{2} T^{5} + 128 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 916 T^{2} + 404806 T^{4} - 916 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 38 T + 722 T^{2} - 19950 T^{3} + 551234 T^{4} - 19950 p^{2} T^{5} + 722 p^{4} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1252 T^{2} + 756838 T^{4} - 1252 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 18 T + 1962 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 60 T + 1800 T^{2} - 30420 T^{3} - 228946 T^{4} - 30420 p^{2} T^{5} + 1800 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 26 T + 210 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 94 T + 4418 T^{2} + 275702 T^{3} + 16029922 T^{4} + 275702 p^{2} T^{5} + 4418 p^{4} T^{6} + 94 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 106 T + 5618 T^{2} + 328706 T^{3} + 18436738 T^{4} + 328706 p^{2} T^{5} + 5618 p^{4} T^{6} + 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7764 T^{2} + 37391750 T^{4} - 7764 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 102 T + 10002 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 126 T + 7938 T^{2} + 813078 T^{3} + 79425122 T^{4} + 813078 p^{2} T^{5} + 7938 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 78 T + 11562 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 148 T + 10952 T^{2} - 999740 T^{3} + 89226574 T^{4} - 999740 p^{2} T^{5} + 10952 p^{4} T^{6} - 148 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 3460 T^{2} + 70145158 T^{4} - 3460 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 186 T + 17298 T^{2} - 1624338 T^{3} + 149130242 T^{4} - 1624338 p^{2} T^{5} + 17298 p^{4} T^{6} - 186 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 23236 T^{2} + 243668806 T^{4} - 23236 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−8.131259100692783560718275386321, −8.074735797722301105134440431940, −7.918678489872580736224189120306, −7.63039277535503686620392038024, −7.37200433438400073662563196652, −6.96385866862724009493926471413, −6.40535558434995312445396326143, −6.40248659197105166701687430530, −6.39365572155687953619730629716, −5.60998194095203382883912012183, −5.52653249666849350016845413860, −5.28147542155698171637176596880, −5.05993834571999006599415088555, −5.01376538111292519735196155129, −4.38461140103571636243720234715, −4.19221984088078723611871303452, −3.50764785323095995067583454776, −3.41647877071700294358497107990, −3.31697240780015625157409397646, −2.56035773165488984279406497033, −2.28116302199687356192786390533, −1.89190981236462948903196991649, −1.71414745324320738938232212414, −0.928350793943685574752278843515, −0.892797942456998278790790021214,
0.892797942456998278790790021214, 0.928350793943685574752278843515, 1.71414745324320738938232212414, 1.89190981236462948903196991649, 2.28116302199687356192786390533, 2.56035773165488984279406497033, 3.31697240780015625157409397646, 3.41647877071700294358497107990, 3.50764785323095995067583454776, 4.19221984088078723611871303452, 4.38461140103571636243720234715, 5.01376538111292519735196155129, 5.05993834571999006599415088555, 5.28147542155698171637176596880, 5.52653249666849350016845413860, 5.60998194095203382883912012183, 6.39365572155687953619730629716, 6.40248659197105166701687430530, 6.40535558434995312445396326143, 6.96385866862724009493926471413, 7.37200433438400073662563196652, 7.63039277535503686620392038024, 7.918678489872580736224189120306, 8.074735797722301105134440431940, 8.131259100692783560718275386321
