| L(s) = 1 | + 0.456·2-s − 1.79·4-s − 0.221·5-s + 4.69·7-s − 1.73·8-s − 0.101·10-s + 11-s + 5.89·13-s + 2.14·14-s + 2.79·16-s − 7.06·17-s + 19-s + 0.397·20-s + 0.456·22-s − 1.06·23-s − 4.95·25-s + 2.68·26-s − 8.41·28-s + 7.62·29-s + 0.901·31-s + 4.73·32-s − 3.22·34-s − 1.04·35-s − 2.71·37-s + 0.456·38-s + 0.384·40-s − 0.788·41-s + ⋯ |
| L(s) = 1 | + 0.322·2-s − 0.895·4-s − 0.0992·5-s + 1.77·7-s − 0.612·8-s − 0.0320·10-s + 0.301·11-s + 1.63·13-s + 0.573·14-s + 0.698·16-s − 1.71·17-s + 0.229·19-s + 0.0889·20-s + 0.0973·22-s − 0.221·23-s − 0.990·25-s + 0.527·26-s − 1.59·28-s + 1.41·29-s + 0.161·31-s + 0.837·32-s − 0.553·34-s − 0.176·35-s − 0.446·37-s + 0.0740·38-s + 0.0607·40-s − 0.123·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.052594417\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.052594417\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.456T + 2T^{2} \) |
| 5 | \( 1 + 0.221T + 5T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 13 | \( 1 - 5.89T + 13T^{2} \) |
| 17 | \( 1 + 7.06T + 17T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 - 7.62T + 29T^{2} \) |
| 31 | \( 1 - 0.901T + 31T^{2} \) |
| 37 | \( 1 + 2.71T + 37T^{2} \) |
| 41 | \( 1 + 0.788T + 41T^{2} \) |
| 43 | \( 1 - 0.714T + 43T^{2} \) |
| 47 | \( 1 + 3.96T + 47T^{2} \) |
| 53 | \( 1 - 9.69T + 53T^{2} \) |
| 59 | \( 1 - 7.33T + 59T^{2} \) |
| 61 | \( 1 - 8.15T + 61T^{2} \) |
| 67 | \( 1 - 7.86T + 67T^{2} \) |
| 71 | \( 1 - 3.13T + 71T^{2} \) |
| 73 | \( 1 + 6.49T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 16.3T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 8.14T + 97T^{2} \) |
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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
−8.829215915730627978853093742860, −8.569055020828761282378686132815, −7.963246964241001909348004629095, −6.75844890292141011402870000762, −5.86432326395574376668530369346, −5.00663021637729751443475590977, −4.31473693523203794741502539808, −3.70335793926854116225750852487, −2.14614216442436132537415853784, −0.997783724846721786686513047855,
0.997783724846721786686513047855, 2.14614216442436132537415853784, 3.70335793926854116225750852487, 4.31473693523203794741502539808, 5.00663021637729751443475590977, 5.86432326395574376668530369346, 6.75844890292141011402870000762, 7.963246964241001909348004629095, 8.569055020828761282378686132815, 8.829215915730627978853093742860
