| L(s) = 1 | + 2·5-s − 9-s + 11-s − 17-s + 19-s + 23-s + 25-s + 4·43-s − 2·45-s − 47-s + 2·55-s − 61-s − 73-s + 2·83-s − 2·85-s + 2·95-s − 99-s − 101-s + 2·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 2·5-s − 9-s + 11-s − 17-s + 19-s + 23-s + 25-s + 4·43-s − 2·45-s − 47-s + 2·55-s − 61-s − 73-s + 2·83-s − 2·85-s + 2·95-s − 99-s − 101-s + 2·115-s + 121-s − 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.269683415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269683415\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.050718350554545922196544990352, −8.828599130541840136746916853701, −8.042345156114895884549894988709, −7.930263798021661188105616645105, −7.17324994670238977183062446602, −7.13627283346170109956270670216, −6.48444124048869285804360487881, −6.16492153340069204519733765910, −5.99129352535923124010317509311, −5.49134708805229452648064660560, −5.43541094583415263882265722168, −4.67641402916326046615712643380, −4.47250342895832001306731680716, −3.82736949648929580698496390120, −3.37487137959198760575014124053, −2.71730143368195118202912078686, −2.55386706215979386477020079443, −2.00689714113842994125695937303, −1.47901243647132623168245217188, −0.918652864142979329308802143353,
0.918652864142979329308802143353, 1.47901243647132623168245217188, 2.00689714113842994125695937303, 2.55386706215979386477020079443, 2.71730143368195118202912078686, 3.37487137959198760575014124053, 3.82736949648929580698496390120, 4.47250342895832001306731680716, 4.67641402916326046615712643380, 5.43541094583415263882265722168, 5.49134708805229452648064660560, 5.99129352535923124010317509311, 6.16492153340069204519733765910, 6.48444124048869285804360487881, 7.13627283346170109956270670216, 7.17324994670238977183062446602, 7.930263798021661188105616645105, 8.042345156114895884549894988709, 8.828599130541840136746916853701, 9.050718350554545922196544990352
