| L(s) = 1 | − 2·2-s − 3-s + 4·4-s + 2·6-s + 10·7-s − 8·8-s − 26·9-s + 11·11-s − 4·12-s + 16·13-s − 20·14-s + 16·16-s − 42·17-s + 52·18-s + 116·19-s − 10·21-s − 22·22-s − 189·23-s + 8·24-s − 32·26-s + 53·27-s + 40·28-s − 120·29-s − 163·31-s − 32·32-s − 11·33-s + 84·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.192·3-s + 1/2·4-s + 0.136·6-s + 0.539·7-s − 0.353·8-s − 0.962·9-s + 0.301·11-s − 0.0962·12-s + 0.341·13-s − 0.381·14-s + 1/4·16-s − 0.599·17-s + 0.680·18-s + 1.40·19-s − 0.103·21-s − 0.213·22-s − 1.71·23-s + 0.0680·24-s − 0.241·26-s + 0.377·27-s + 0.269·28-s − 0.768·29-s − 0.944·31-s − 0.176·32-s − 0.0580·33-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 13 | \( 1 - 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 189 T + p^{3} T^{2} \) |
| 29 | \( 1 + 120 T + p^{3} T^{2} \) |
| 31 | \( 1 + 163 T + p^{3} T^{2} \) |
| 37 | \( 1 - 409 T + p^{3} T^{2} \) |
| 41 | \( 1 - 468 T + p^{3} T^{2} \) |
| 43 | \( 1 + 110 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 + 90 T + p^{3} T^{2} \) |
| 59 | \( 1 + 453 T + p^{3} T^{2} \) |
| 61 | \( 1 - 20 T + p^{3} T^{2} \) |
| 67 | \( 1 - 97 T + p^{3} T^{2} \) |
| 71 | \( 1 + 465 T + p^{3} T^{2} \) |
| 73 | \( 1 + 848 T + p^{3} T^{2} \) |
| 79 | \( 1 + 742 T + p^{3} T^{2} \) |
| 83 | \( 1 + 438 T + p^{3} T^{2} \) |
| 89 | \( 1 + 273 T + p^{3} T^{2} \) |
| 97 | \( 1 + 761 T + p^{3} T^{2} \) |
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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
−9.781490440194465031761464399418, −9.118041139190887783756742623834, −8.139922817075211611891164600254, −7.50754366206605361090710500222, −6.21800635108522908708199851194, −5.53721567368274676935955976887, −4.13439774818370383725110499228, −2.78033951042288490809570261059, −1.47332427658300395880352447823, 0,
1.47332427658300395880352447823, 2.78033951042288490809570261059, 4.13439774818370383725110499228, 5.53721567368274676935955976887, 6.21800635108522908708199851194, 7.50754366206605361090710500222, 8.139922817075211611891164600254, 9.118041139190887783756742623834, 9.781490440194465031761464399418
