| L(s) = 1 | − 2-s + 5·7-s + 8-s + 11-s − 2·13-s − 5·14-s − 16-s − 17-s − 5·19-s − 22-s + 2·23-s + 5·25-s + 2·26-s + 10·29-s + 8·31-s + 34-s + 8·37-s + 5·38-s − 12·43-s − 2·46-s + 11·47-s + 18·49-s − 5·50-s + 5·53-s + 5·56-s − 10·58-s − 3·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.88·7-s + 0.353·8-s + 0.301·11-s − 0.554·13-s − 1.33·14-s − 1/4·16-s − 0.242·17-s − 1.14·19-s − 0.213·22-s + 0.417·23-s + 25-s + 0.392·26-s + 1.85·29-s + 1.43·31-s + 0.171·34-s + 1.31·37-s + 0.811·38-s − 1.82·43-s − 0.294·46-s + 1.60·47-s + 18/7·49-s − 0.707·50-s + 0.686·53-s + 0.668·56-s − 1.31·58-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.332042497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.332042497\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 8 T + 27 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 11 T + 74 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 16 T + 167 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.324212620875756592404289019577, −9.290163571141235009345698553588, −8.569669569296550001837164432766, −8.416638518301415063244384139027, −8.052407075432851372862965641755, −7.951759678861771154663020116901, −7.13686611145762101365056805473, −6.93811517446882385959504478452, −6.47058107421488531061756114574, −6.05151223945454954642411532183, −5.34779681788081531562185630051, −4.83946770356458674943453839329, −4.72936903056450996255081344261, −4.34744571534578184053435646016, −3.75075434047520526625475754431, −2.96314337780601558094647901632, −2.24197978126696964981295894954, −2.15243148971300611506108088343, −1.05437542937507547113199125490, −0.855803606293805256297140439447,
0.855803606293805256297140439447, 1.05437542937507547113199125490, 2.15243148971300611506108088343, 2.24197978126696964981295894954, 2.96314337780601558094647901632, 3.75075434047520526625475754431, 4.34744571534578184053435646016, 4.72936903056450996255081344261, 4.83946770356458674943453839329, 5.34779681788081531562185630051, 6.05151223945454954642411532183, 6.47058107421488531061756114574, 6.93811517446882385959504478452, 7.13686611145762101365056805473, 7.951759678861771154663020116901, 8.052407075432851372862965641755, 8.416638518301415063244384139027, 8.569669569296550001837164432766, 9.290163571141235009345698553588, 9.324212620875756592404289019577
