| L(s) = 1 | − 2.06·2-s − 2.69·3-s + 2.27·4-s + 4.37·5-s + 5.57·6-s + 3.31·7-s − 0.561·8-s + 4.28·9-s − 9.03·10-s + 3.63·11-s − 6.13·12-s + 6.48·13-s − 6.85·14-s − 11.7·15-s − 3.38·16-s − 0.501·17-s − 8.85·18-s − 4.21·19-s + 9.93·20-s − 8.94·21-s − 7.50·22-s − 0.270·23-s + 1.51·24-s + 14.1·25-s − 13.4·26-s − 3.46·27-s + 7.53·28-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 1.55·3-s + 1.13·4-s + 1.95·5-s + 2.27·6-s + 1.25·7-s − 0.198·8-s + 1.42·9-s − 2.85·10-s + 1.09·11-s − 1.76·12-s + 1.79·13-s − 1.83·14-s − 3.04·15-s − 0.845·16-s − 0.121·17-s − 2.08·18-s − 0.966·19-s + 2.22·20-s − 1.95·21-s − 1.60·22-s − 0.0564·23-s + 0.309·24-s + 2.82·25-s − 2.62·26-s − 0.666·27-s + 1.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7799699961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7799699961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 - T \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 4.37T + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 3.63T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 + 0.501T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 + 0.270T + 23T^{2} \) |
| 29 | \( 1 + 0.256T + 29T^{2} \) |
| 31 | \( 1 + 2.66T + 31T^{2} \) |
| 37 | \( 1 + 4.33T + 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 + 8.53T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 8.69T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 0.162T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 + 3.99T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 4.96T + 79T^{2} \) |
| 83 | \( 1 - 5.07T + 83T^{2} \) |
| 89 | \( 1 + 5.28T + 89T^{2} \) |
| 97 | \( 1 + 5.98T + 97T^{2} \) |
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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
−10.70457567475475995489030566642, −10.13174498992407191395192543544, −8.967945497103319701614336851652, −8.601661350383446971543576355740, −6.98064489416902508650216352685, −6.27231002829127168066775580337, −5.62045613926575853834223163548, −4.49775516580591863051783365452, −1.73201786461781277174456265486, −1.28302944083178864723734777969,
1.28302944083178864723734777969, 1.73201786461781277174456265486, 4.49775516580591863051783365452, 5.62045613926575853834223163548, 6.27231002829127168066775580337, 6.98064489416902508650216352685, 8.601661350383446971543576355740, 8.967945497103319701614336851652, 10.13174498992407191395192543544, 10.70457567475475995489030566642
