| L(s) = 1 | + 16-s − 2·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-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 | + 16-s − 2·25-s + 4·67-s − 4·79-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8691068670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8691068670\) |
| \(L(1)\) |
|
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 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.11043012879546794792683810892, −5.94829828174593325504806339157, −5.74277810231563740501728061195, −5.62118330665145066102425839919, −5.38080501808663299775779592343, −5.20633441695031926336165519034, −5.01174706373990138068642246131, −4.83046088358181510458449318977, −4.67162470091720355594567084568, −4.20968621606679618683479706587, −4.09762569699919628649442285259, −3.87956755178082820975356621848, −3.72032912777597919451117200989, −3.69995649107587175395581497799, −3.40225787720413641132118707418, −2.96212317648039465925110848689, −2.78466678808260653226199162778, −2.65480485602388637514979764527, −2.26912644634595336716858864484, −2.20917855984096276975244082780, −1.85598555304591033844870426382, −1.35139518479425626988002304017, −1.29770229535788770028691411624, −1.12187769223755682377893871922, −0.32288522951787767770156516364,
0.32288522951787767770156516364, 1.12187769223755682377893871922, 1.29770229535788770028691411624, 1.35139518479425626988002304017, 1.85598555304591033844870426382, 2.20917855984096276975244082780, 2.26912644634595336716858864484, 2.65480485602388637514979764527, 2.78466678808260653226199162778, 2.96212317648039465925110848689, 3.40225787720413641132118707418, 3.69995649107587175395581497799, 3.72032912777597919451117200989, 3.87956755178082820975356621848, 4.09762569699919628649442285259, 4.20968621606679618683479706587, 4.67162470091720355594567084568, 4.83046088358181510458449318977, 5.01174706373990138068642246131, 5.20633441695031926336165519034, 5.38080501808663299775779592343, 5.62118330665145066102425839919, 5.74277810231563740501728061195, 5.94829828174593325504806339157, 6.11043012879546794792683810892
