| L(s) = 1 | + 2·5-s + 2·7-s − 2·9-s + 11-s + 2·19-s + 25-s + 4·35-s + 2·43-s − 4·45-s + 49-s + 2·55-s − 4·63-s + 2·77-s + 3·81-s − 4·83-s + 4·95-s − 2·99-s − 2·125-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·5-s + 2·7-s − 2·9-s + 11-s + 2·19-s + 25-s + 4·35-s + 2·43-s − 4·45-s + 49-s + 2·55-s − 4·63-s + 2·77-s + 3·81-s − 4·83-s + 4·95-s − 2·99-s − 2·125-s + 127-s + 131-s + 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11182336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.764727914\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.764727914\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.993804142314755976768645742234, −8.662696738312860166938958859823, −8.301059966152248359103091616081, −7.990437356527933320210267153483, −7.39862789946841542544627402278, −7.38575284975012575810355834059, −6.53981517395737932477705907068, −6.27372223994604475169291171305, −5.78975655631155232458125327450, −5.61658216405953371288438319836, −5.26314336614174830622068832088, −5.13432971017195760823331787834, −4.35642101593796459106114158483, −4.11923576692080355485624399210, −3.22183988221379170375321930166, −3.01677768875011521869734197634, −2.27417374635565346642232489292, −2.14551292080394470732647440931, −1.29148352670585493178495700304, −1.25670040207136855628877895273,
1.25670040207136855628877895273, 1.29148352670585493178495700304, 2.14551292080394470732647440931, 2.27417374635565346642232489292, 3.01677768875011521869734197634, 3.22183988221379170375321930166, 4.11923576692080355485624399210, 4.35642101593796459106114158483, 5.13432971017195760823331787834, 5.26314336614174830622068832088, 5.61658216405953371288438319836, 5.78975655631155232458125327450, 6.27372223994604475169291171305, 6.53981517395737932477705907068, 7.38575284975012575810355834059, 7.39862789946841542544627402278, 7.990437356527933320210267153483, 8.301059966152248359103091616081, 8.662696738312860166938958859823, 8.993804142314755976768645742234
