| L(s) = 1 | − 2.74·2-s − 2.10·3-s + 5.55·4-s − 2.44·5-s + 5.77·6-s − 9.76·8-s + 1.42·9-s + 6.71·10-s + 11-s − 11.6·12-s − 1.89·13-s + 5.13·15-s + 15.7·16-s − 3.95·17-s − 3.90·18-s − 2.88·19-s − 13.5·20-s − 2.74·22-s − 6.59·23-s + 20.5·24-s + 0.970·25-s + 5.19·26-s + 3.31·27-s − 0.675·29-s − 14.1·30-s − 7.81·31-s − 23.6·32-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 1.21·3-s + 2.77·4-s − 1.09·5-s + 2.35·6-s − 3.45·8-s + 0.473·9-s + 2.12·10-s + 0.301·11-s − 3.37·12-s − 0.524·13-s + 1.32·15-s + 3.93·16-s − 0.958·17-s − 0.920·18-s − 0.662·19-s − 3.03·20-s − 0.585·22-s − 1.37·23-s + 4.19·24-s + 0.194·25-s + 1.01·26-s + 0.638·27-s − 0.125·29-s − 2.57·30-s − 1.40·31-s − 4.18·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1565335880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1565335880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 + 2.10T + 3T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 13 | \( 1 + 1.89T + 13T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 + 0.675T + 29T^{2} \) |
| 31 | \( 1 + 7.81T + 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 - 3.26T + 41T^{2} \) |
| 43 | \( 1 + 1.06T + 43T^{2} \) |
| 47 | \( 1 - 9.70T + 47T^{2} \) |
| 53 | \( 1 - 3.74T + 53T^{2} \) |
| 59 | \( 1 - 6.48T + 59T^{2} \) |
| 61 | \( 1 - 6.09T + 61T^{2} \) |
| 67 | \( 1 - 4.39T + 67T^{2} \) |
| 71 | \( 1 - 8.39T + 71T^{2} \) |
| 73 | \( 1 + 2.66T + 73T^{2} \) |
| 79 | \( 1 - 2.09T + 79T^{2} \) |
| 83 | \( 1 + 3.78T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.86971405696313286427850118300, −10.01795278138125637296060094237, −9.031643678685241908038167815312, −8.191845955493922428236095844894, −7.36591418270637121648833747676, −6.60571534508742915583465487480, −5.70377652318891225446830237804, −4.03463411177253111473077227838, −2.23646708287444702920912193916, −0.47087446010491116405281159648,
0.47087446010491116405281159648, 2.23646708287444702920912193916, 4.03463411177253111473077227838, 5.70377652318891225446830237804, 6.60571534508742915583465487480, 7.36591418270637121648833747676, 8.191845955493922428236095844894, 9.031643678685241908038167815312, 10.01795278138125637296060094237, 10.86971405696313286427850118300
