| L(s) = 1 | + 2·3-s − 14·7-s + 2·9-s − 20·11-s − 18·13-s − 2·17-s − 28·21-s − 46·23-s + 18·27-s + 28·31-s − 40·33-s − 66·37-s − 36·39-s − 28·41-s − 30·43-s − 78·47-s + 98·49-s − 4·51-s + 14·53-s + 84·61-s − 28·63-s − 14·67-s − 92·69-s − 196·71-s − 98·73-s + 280·77-s − 13·81-s + ⋯ |
| L(s) = 1 | + 2/3·3-s − 2·7-s + 2/9·9-s − 1.81·11-s − 1.38·13-s − 0.117·17-s − 4/3·21-s − 2·23-s + 2/3·27-s + 0.903·31-s − 1.21·33-s − 1.78·37-s − 0.923·39-s − 0.682·41-s − 0.697·43-s − 1.65·47-s + 2·49-s − 0.0784·51-s + 0.264·53-s + 1.37·61-s − 4/9·63-s − 0.208·67-s − 4/3·69-s − 2.76·71-s − 1.34·73-s + 3.63·77-s − 0.160·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03517703067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03517703067\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 658 T^{2} + p^{4} T^{4} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 1618 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 66 T + 2178 T^{2} + 66 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 30 T + 450 T^{2} + 30 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 78 T + 3042 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3826 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 98 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 98 T + 4802 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3266 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3298 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 66 T + 2178 T^{2} + 66 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.41499513159046605609880188291, −10.34075574639242689201889643153, −10.22727968251262811586009858664, −9.924881399325737445108897709911, −9.848243468833479780588358365579, −8.896855359243206894741192457438, −8.572085299272496551984124415464, −8.120787698760519666875659126330, −7.54623056790744008073582091156, −7.03152782655453350323850298444, −6.77842899970267672950284916551, −5.90458906093560965323063219196, −5.72744098947145072525660119313, −4.80020209566613135711326567837, −4.48457765401652588862314413602, −3.30990526953465270386917161561, −3.26555131103176463657106291736, −2.53956892720900427113331112727, −1.96140755417840218347593722656, −0.07153432332835254383280210600,
0.07153432332835254383280210600, 1.96140755417840218347593722656, 2.53956892720900427113331112727, 3.26555131103176463657106291736, 3.30990526953465270386917161561, 4.48457765401652588862314413602, 4.80020209566613135711326567837, 5.72744098947145072525660119313, 5.90458906093560965323063219196, 6.77842899970267672950284916551, 7.03152782655453350323850298444, 7.54623056790744008073582091156, 8.120787698760519666875659126330, 8.572085299272496551984124415464, 8.896855359243206894741192457438, 9.848243468833479780588358365579, 9.924881399325737445108897709911, 10.22727968251262811586009858664, 10.34075574639242689201889643153, 11.41499513159046605609880188291
