| L(s) = 1 | + 5-s − 9-s + 11-s + 17-s − 19-s − 2·23-s + 25-s − 2·43-s − 45-s + 47-s + 55-s + 61-s + 73-s + 4·83-s + 85-s − 95-s − 99-s − 2·101-s − 2·115-s + 121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 5-s − 9-s + 11-s + 17-s − 19-s − 2·23-s + 25-s − 2·43-s − 45-s + 47-s + 55-s + 61-s + 73-s + 4·83-s + 85-s − 95-s − 99-s − 2·101-s − 2·115-s + 121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13868176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.595807069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595807069\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$ | \( ( 1 - T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−8.717109917227330457579181591885, −8.699629564000920235757333711119, −8.124982455574306957142741798460, −7.909236072443285415643004601586, −7.50783802004575073445301186780, −6.81166801410978967905108111328, −6.48957887246313796201305956739, −6.43768228897130868959087639998, −5.85600528650101313618957763664, −5.61881413269342878491488954252, −5.23116276040291331517536910185, −4.81419628247920123479889106032, −4.23782174999867611170028495022, −3.81565440144836153143760433219, −3.47791715399893294117677873613, −2.97264276023836715606033146379, −2.34453309674846738681748061847, −2.01209835935887637317845673145, −1.56922280159680543184160473022, −0.72593903728398656606338243250,
0.72593903728398656606338243250, 1.56922280159680543184160473022, 2.01209835935887637317845673145, 2.34453309674846738681748061847, 2.97264276023836715606033146379, 3.47791715399893294117677873613, 3.81565440144836153143760433219, 4.23782174999867611170028495022, 4.81419628247920123479889106032, 5.23116276040291331517536910185, 5.61881413269342878491488954252, 5.85600528650101313618957763664, 6.43768228897130868959087639998, 6.48957887246313796201305956739, 6.81166801410978967905108111328, 7.50783802004575073445301186780, 7.909236072443285415643004601586, 8.124982455574306957142741798460, 8.699629564000920235757333711119, 8.717109917227330457579181591885
