| L(s) = 1 | − 4-s + 2·9-s + 16-s + 2·17-s + 25-s − 2·29-s − 2·36-s − 2·49-s − 2·53-s − 2·61-s − 64-s − 2·68-s + 3·81-s − 100-s + 2·101-s − 2·113-s + 2·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 4·153-s + 157-s + ⋯ |
| L(s) = 1 | − 4-s + 2·9-s + 16-s + 2·17-s + 25-s − 2·29-s − 2·36-s − 2·49-s − 2·53-s − 2·61-s − 64-s − 2·68-s + 3·81-s − 100-s + 2·101-s − 2·113-s + 2·116-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 4·153-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8327570625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8327570625\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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
−10.69108350503234792112174303004, −10.46870693552330568644087118685, −9.906183274459220138093506176539, −9.658202526604749882294129989996, −9.311721818619462828716432241233, −9.001450219269808719004164924414, −8.049231277810955034779837119765, −8.007991091956880870663342401106, −7.38639700795928015494000202826, −7.26072005063231472595606729850, −6.36258253437526093266638341539, −6.10443864946433196507472156121, −5.31426646929877387509070449192, −4.91554474485232945717649467416, −4.62324686760569529842796746024, −3.81464171922524960755544021108, −3.59096776911207756611741946119, −2.95028313491531740824300178130, −1.62826427500627072966391379375, −1.32004426888021298833398603327,
1.32004426888021298833398603327, 1.62826427500627072966391379375, 2.95028313491531740824300178130, 3.59096776911207756611741946119, 3.81464171922524960755544021108, 4.62324686760569529842796746024, 4.91554474485232945717649467416, 5.31426646929877387509070449192, 6.10443864946433196507472156121, 6.36258253437526093266638341539, 7.26072005063231472595606729850, 7.38639700795928015494000202826, 8.007991091956880870663342401106, 8.049231277810955034779837119765, 9.001450219269808719004164924414, 9.311721818619462828716432241233, 9.658202526604749882294129989996, 9.906183274459220138093506176539, 10.46870693552330568644087118685, 10.69108350503234792112174303004
