| L(s) = 1 | + 2·2-s + 4·4-s + 3·5-s + 10·7-s + 8·8-s + 6·10-s + 16·13-s + 20·14-s + 16·16-s + 42·17-s − 116·19-s + 12·20-s − 189·23-s − 116·25-s + 32·26-s + 40·28-s − 120·29-s − 163·31-s + 32·32-s + 84·34-s + 30·35-s − 409·37-s − 232·38-s + 24·40-s + 468·41-s − 110·43-s − 378·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.268·5-s + 0.539·7-s + 0.353·8-s + 0.189·10-s + 0.341·13-s + 0.381·14-s + 1/4·16-s + 0.599·17-s − 1.40·19-s + 0.134·20-s − 1.71·23-s − 0.927·25-s + 0.241·26-s + 0.269·28-s − 0.768·29-s − 0.944·31-s + 0.176·32-s + 0.423·34-s + 0.144·35-s − 1.81·37-s − 0.990·38-s + 0.0948·40-s + 1.78·41-s − 0.390·43-s − 1.21·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p^{3} T^{2} \) |
| 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
−8.150740318197506828594077050116, −7.59980823537203365051032639281, −6.54356363138784891783220121532, −5.87327658092856678161189384159, −5.20170444139887102195770931559, −4.14709232831780207121872975870, −3.60369871349062558554019239260, −2.22637137268872572417726684903, −1.64004604501490773613050712896, 0,
1.64004604501490773613050712896, 2.22637137268872572417726684903, 3.60369871349062558554019239260, 4.14709232831780207121872975870, 5.20170444139887102195770931559, 5.87327658092856678161189384159, 6.54356363138784891783220121532, 7.59980823537203365051032639281, 8.150740318197506828594077050116
