| L(s) = 1 | + 2-s − 3-s − 6-s − 8-s − 3·11-s − 16-s + 19-s − 3·22-s + 24-s + 25-s + 27-s + 3·33-s + 38-s + 41-s − 43-s + 48-s − 49-s + 50-s + 54-s − 57-s − 59-s + 64-s + 3·66-s − 3·67-s + 2·73-s − 75-s − 81-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 6-s − 8-s − 3·11-s − 16-s + 19-s − 3·22-s + 24-s + 25-s + 27-s + 3·33-s + 38-s + 41-s − 43-s + 48-s − 49-s + 50-s + 54-s − 57-s − 59-s + 64-s + 3·66-s − 3·67-s + 2·73-s − 75-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6086119383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6086119383\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$ | \( ( 1 + T )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−10.19992407014213507329314893543, −9.742837198637934637974034440040, −9.043044991540750818265817362150, −8.953728124224133310576321510780, −8.110951552174974471932237545516, −8.002279276761307831718260393146, −7.63670378739673953381509578231, −6.83068697769412381049165946727, −6.78924731069484555945057345848, −5.86494250506469358928852376920, −5.77333008573524034871407430954, −5.38045687324001764086945309329, −5.04881230319214055912586177595, −4.56571333109753388850040806957, −4.40955263132450233428023203095, −3.15647844675266913186141739174, −3.14820470194036311902705481718, −2.73257050027175518378664748740, −1.88196372475311117823275748292, −0.57279851338610339336732845235,
0.57279851338610339336732845235, 1.88196372475311117823275748292, 2.73257050027175518378664748740, 3.14820470194036311902705481718, 3.15647844675266913186141739174, 4.40955263132450233428023203095, 4.56571333109753388850040806957, 5.04881230319214055912586177595, 5.38045687324001764086945309329, 5.77333008573524034871407430954, 5.86494250506469358928852376920, 6.78924731069484555945057345848, 6.83068697769412381049165946727, 7.63670378739673953381509578231, 8.002279276761307831718260393146, 8.110951552174974471932237545516, 8.953728124224133310576321510780, 9.043044991540750818265817362150, 9.742837198637934637974034440040, 10.19992407014213507329314893543
