| L(s) = 1 | + 2-s + 3·5-s − 7-s − 8-s + 3·10-s − 11-s + 4·13-s − 14-s − 16-s − 3·17-s − 2·19-s − 22-s + 3·23-s + 5·25-s + 4·26-s − 2·31-s − 3·34-s − 3·35-s − 8·37-s − 2·38-s − 3·40-s + 18·41-s − 8·43-s + 3·46-s + 3·47-s − 6·49-s + 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s − 0.301·11-s + 1.10·13-s − 0.267·14-s − 1/4·16-s − 0.727·17-s − 0.458·19-s − 0.213·22-s + 0.625·23-s + 25-s + 0.784·26-s − 0.359·31-s − 0.514·34-s − 0.507·35-s − 1.31·37-s − 0.324·38-s − 0.474·40-s + 2.81·41-s − 1.21·43-s + 0.442·46-s + 0.437·47-s − 6/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.581144131\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.581144131\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−9.833303178867000007867976614648, −9.217062924596311248676893253909, −9.076758327552317351774415152955, −8.697821537925851870849946695263, −8.333286601262976603405087298477, −7.66175022574614591161562373478, −7.26197844286975942477499885528, −6.57120440238752923490264835571, −6.55230463146272332999433292479, −5.92490155316792671113020301215, −5.72236823786341076246192069360, −5.22372394999525014477091871357, −4.81131089079268160735577534770, −4.15138549989586915671663184457, −3.92243425725762980072636494061, −3.02519036030764978618630354830, −2.91876392780964642375102573307, −2.05041342205295863930666685506, −1.68012959672932674705213417658, −0.68325844302784036190369189867,
0.68325844302784036190369189867, 1.68012959672932674705213417658, 2.05041342205295863930666685506, 2.91876392780964642375102573307, 3.02519036030764978618630354830, 3.92243425725762980072636494061, 4.15138549989586915671663184457, 4.81131089079268160735577534770, 5.22372394999525014477091871357, 5.72236823786341076246192069360, 5.92490155316792671113020301215, 6.55230463146272332999433292479, 6.57120440238752923490264835571, 7.26197844286975942477499885528, 7.66175022574614591161562373478, 8.333286601262976603405087298477, 8.697821537925851870849946695263, 9.076758327552317351774415152955, 9.217062924596311248676893253909, 9.833303178867000007867976614648
