| L(s) = 1 | + 5.98e3·16-s + 6.25e4·25-s + 2.15e5·67-s + 3.71e6·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.93e7·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 1.46·16-s + 4·25-s + 0.717·67-s + 7.53·79-s − 4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(9.016587083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.016587083\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 5983 T^{4} + p^{24} T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 11 | $C_2^3$ | \( 1 - 6183023585758 T^{4} + p^{24} T^{8} \) |
| 13 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 4907043591387842 T^{4} + p^{24} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 161104468562341918 T^{4} + p^{24} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 5108772818 T^{2} + p^{12} T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3388378898 T^{2} + p^{12} T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 53 | $C_2^3$ | \( 1 + \)\(76\!\cdots\!82\)\( T^{4} + p^{24} T^{8} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 53926 T + p^{6} T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 + \)\(61\!\cdots\!82\)\( T^{4} + p^{24} T^{8} \) |
| 73 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 929378 T + p^{6} T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{4}( 1 + p^{3} T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{4} \) |
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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
−7.18422811778529935442572068653, −6.60684758489595526663176487965, −6.51269656975138188211243221214, −6.34750421915731901522792309501, −6.23722435632864315429698735538, −5.73591529372381699797057966105, −5.28929592276057384098839443123, −5.23556180686404266210742028250, −5.13005379572811586836439290406, −4.75064457191277453525713439690, −4.64367630583306731908018394870, −4.07568624223204238990916426308, −3.94099184976115612937135397434, −3.55435677212633233945234024435, −3.22190568387535260920457719597, −3.14595129882137540185181617874, −2.89998785218860511647655278333, −2.33495180994586547276611234722, −2.23398721382805402051870178782, −1.87577923238335804726227898178, −1.39965543654658930638383608683, −0.954375291428886730553209119868, −0.904822963802433402650403273195, −0.72898344072074553973700614677, −0.29781369400408181886143176850,
0.29781369400408181886143176850, 0.72898344072074553973700614677, 0.904822963802433402650403273195, 0.954375291428886730553209119868, 1.39965543654658930638383608683, 1.87577923238335804726227898178, 2.23398721382805402051870178782, 2.33495180994586547276611234722, 2.89998785218860511647655278333, 3.14595129882137540185181617874, 3.22190568387535260920457719597, 3.55435677212633233945234024435, 3.94099184976115612937135397434, 4.07568624223204238990916426308, 4.64367630583306731908018394870, 4.75064457191277453525713439690, 5.13005379572811586836439290406, 5.23556180686404266210742028250, 5.28929592276057384098839443123, 5.73591529372381699797057966105, 6.23722435632864315429698735538, 6.34750421915731901522792309501, 6.51269656975138188211243221214, 6.60684758489595526663176487965, 7.18422811778529935442572068653
