| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 2·13-s + 16-s − 25-s − 4·27-s − 3·36-s − 4·39-s − 4·43-s − 2·48-s − 2·49-s − 2·52-s − 64-s + 2·75-s + 4·79-s + 5·81-s + 100-s + 4·108-s + 6·117-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 2·3-s − 4-s + 3·9-s + 2·12-s + 2·13-s + 16-s − 25-s − 4·27-s − 3·36-s − 4·39-s − 4·43-s − 2·48-s − 2·49-s − 2·52-s − 64-s + 2·75-s + 4·79-s + 5·81-s + 100-s + 4·108-s + 6·117-s − 2·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4059059035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4059059035\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$ | \( ( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{4} \) |
| 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
−9.891767780930705834507708458391, −9.465881515785681660784069950311, −9.280415348874297448936626358500, −8.570554956129462779228584161184, −8.076897121303921053203726152889, −8.047314160610133101530080005359, −7.41310843047231011074407866626, −6.61618216905513338139999218469, −6.49147006486127271139307636863, −6.31658448585394989182106715003, −5.65791256348840299180803701653, −5.17028030406533126560967941397, −5.14637900411488151018799580618, −4.47929764363723737965942924075, −4.05881082455783942215491415621, −3.51548440099056782291667955892, −3.33109429810329716051787720410, −1.72960181570808695176212245361, −1.60489032569598657217205765218, −0.62866525652835745724448548805,
0.62866525652835745724448548805, 1.60489032569598657217205765218, 1.72960181570808695176212245361, 3.33109429810329716051787720410, 3.51548440099056782291667955892, 4.05881082455783942215491415621, 4.47929764363723737965942924075, 5.14637900411488151018799580618, 5.17028030406533126560967941397, 5.65791256348840299180803701653, 6.31658448585394989182106715003, 6.49147006486127271139307636863, 6.61618216905513338139999218469, 7.41310843047231011074407866626, 8.047314160610133101530080005359, 8.076897121303921053203726152889, 8.570554956129462779228584161184, 9.280415348874297448936626358500, 9.465881515785681660784069950311, 9.891767780930705834507708458391
