| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s − 8·6-s − 20·7-s + 8·8-s − 11·9-s + 11·11-s − 16·12-s − 26·13-s − 40·14-s + 16·16-s + 42·17-s − 22·18-s + 116·19-s + 80·21-s + 22·22-s − 96·23-s − 32·24-s − 52·26-s + 152·27-s − 80·28-s + 270·29-s + 32·31-s + 32·32-s − 44·33-s + 84·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.544·6-s − 1.07·7-s + 0.353·8-s − 0.407·9-s + 0.301·11-s − 0.384·12-s − 0.554·13-s − 0.763·14-s + 1/4·16-s + 0.599·17-s − 0.288·18-s + 1.40·19-s + 0.831·21-s + 0.213·22-s − 0.870·23-s − 0.272·24-s − 0.392·26-s + 1.08·27-s − 0.539·28-s + 1.72·29-s + 0.185·31-s + 0.176·32-s − 0.232·33-s + 0.423·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.855267160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855267160\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 270 T + p^{3} T^{2} \) |
| 31 | \( 1 - 32 T + p^{3} T^{2} \) |
| 37 | \( 1 - 106 T + p^{3} T^{2} \) |
| 41 | \( 1 + 462 T + p^{3} T^{2} \) |
| 43 | \( 1 - 40 T + p^{3} T^{2} \) |
| 47 | \( 1 - 504 T + p^{3} T^{2} \) |
| 53 | \( 1 - 570 T + p^{3} T^{2} \) |
| 59 | \( 1 - 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 590 T + p^{3} T^{2} \) |
| 67 | \( 1 - 388 T + p^{3} T^{2} \) |
| 71 | \( 1 + 240 T + p^{3} T^{2} \) |
| 73 | \( 1 + 302 T + p^{3} T^{2} \) |
| 79 | \( 1 - 8 T + p^{3} T^{2} \) |
| 83 | \( 1 - 48 T + p^{3} T^{2} \) |
| 89 | \( 1 - 282 T + p^{3} T^{2} \) |
| 97 | \( 1 - 646 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.32002782409273336596716402898, −9.870833230541643391923256351147, −8.603790594725400416938942686860, −7.35692158441482337864049646292, −6.49350711618721320741673438405, −5.75135730552851627835460700403, −4.92216677344571270966857252006, −3.61216206643723181482269141439, −2.66873786900275204770967184638, −0.76402215059878645310894876614,
0.76402215059878645310894876614, 2.66873786900275204770967184638, 3.61216206643723181482269141439, 4.92216677344571270966857252006, 5.75135730552851627835460700403, 6.49350711618721320741673438405, 7.35692158441482337864049646292, 8.603790594725400416938942686860, 9.870833230541643391923256351147, 10.32002782409273336596716402898
