| L(s) = 1 | − 2-s + 2·5-s + 7-s + 8-s − 2·10-s + 4·11-s − 6·13-s − 14-s − 16-s + 4·17-s − 8·19-s − 4·22-s − 8·23-s + 5·25-s + 6·26-s + 2·29-s − 4·34-s + 2·35-s − 20·37-s + 8·38-s + 2·40-s + 6·41-s + 4·43-s + 8·46-s − 5·50-s + 12·53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.894·5-s + 0.377·7-s + 0.353·8-s − 0.632·10-s + 1.20·11-s − 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.970·17-s − 1.83·19-s − 0.852·22-s − 1.66·23-s + 25-s + 1.17·26-s + 0.371·29-s − 0.685·34-s + 0.338·35-s − 3.28·37-s + 1.29·38-s + 0.316·40-s + 0.937·41-s + 0.609·43-s + 1.17·46-s − 0.707·50-s + 1.64·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.342081669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.342081669\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.885778667519538311425585998505, −9.613628234917381576404022668893, −9.345475652651039134221630359466, −8.669388729820317307964743941555, −8.571211596348526381669538597836, −8.143009293789829425246102601709, −7.49471232616844452988196306763, −7.20578254342528921001303957977, −6.71890543664226226781911190105, −6.35931100083558245288351769371, −5.78327487432759317661103185469, −5.45594299134574925731759233087, −4.67956484641865475321130999341, −4.63157827510658929136200378650, −3.78792042335503094939221623224, −3.44220668104326879194820005947, −2.32742428023720118268291534390, −2.14861667714128077134085557197, −1.56740343359640761675872133826, −0.57246189713684143371210172076,
0.57246189713684143371210172076, 1.56740343359640761675872133826, 2.14861667714128077134085557197, 2.32742428023720118268291534390, 3.44220668104326879194820005947, 3.78792042335503094939221623224, 4.63157827510658929136200378650, 4.67956484641865475321130999341, 5.45594299134574925731759233087, 5.78327487432759317661103185469, 6.35931100083558245288351769371, 6.71890543664226226781911190105, 7.20578254342528921001303957977, 7.49471232616844452988196306763, 8.143009293789829425246102601709, 8.571211596348526381669538597836, 8.669388729820317307964743941555, 9.345475652651039134221630359466, 9.613628234917381576404022668893, 9.885778667519538311425585998505
