| L(s) = 1 | + 2·3-s − 3·9-s − 14·27-s − 8·31-s + 8·37-s + 6·41-s − 16·43-s − 2·49-s + 24·53-s + 18·67-s + 32·71-s + 8·79-s − 4·81-s − 2·83-s − 26·89-s − 16·93-s + 26·107-s + 16·111-s + 13·121-s + 12·123-s + 127-s − 32·129-s + 131-s + 137-s + 139-s − 4·147-s + 149-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 9-s − 2.69·27-s − 1.43·31-s + 1.31·37-s + 0.937·41-s − 2.43·43-s − 2/7·49-s + 3.29·53-s + 2.19·67-s + 3.79·71-s + 0.900·79-s − 4/9·81-s − 0.219·83-s − 2.75·89-s − 1.65·93-s + 2.51·107-s + 1.51·111-s + 1.18·121-s + 1.08·123-s + 0.0887·127-s − 2.81·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.329·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.391647268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391647268\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 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
−8.696438118881425273346399002281, −8.478682693682227360986449381633, −8.155670886314316423948009514420, −8.012570903099387420093175593332, −7.39402164892776952434994347599, −7.00776740855732643635488433377, −6.80412177784723650676077403747, −6.10803053616986082554035973017, −5.75169458892896113328667769082, −5.58289979669047444827529227913, −4.93208177667144126118351092043, −4.74946690227103099310791070625, −3.76460557451221756239824423097, −3.59438685322385097325814238541, −3.56858947763654977949171364857, −2.62794821220918021775694594016, −2.31919723744166946149789178290, −2.20137801989844879994723769684, −1.22902502393988523545545152192, −0.44637383891249922654298228084,
0.44637383891249922654298228084, 1.22902502393988523545545152192, 2.20137801989844879994723769684, 2.31919723744166946149789178290, 2.62794821220918021775694594016, 3.56858947763654977949171364857, 3.59438685322385097325814238541, 3.76460557451221756239824423097, 4.74946690227103099310791070625, 4.93208177667144126118351092043, 5.58289979669047444827529227913, 5.75169458892896113328667769082, 6.10803053616986082554035973017, 6.80412177784723650676077403747, 7.00776740855732643635488433377, 7.39402164892776952434994347599, 8.012570903099387420093175593332, 8.155670886314316423948009514420, 8.478682693682227360986449381633, 8.696438118881425273346399002281
