| L(s) = 1 | + 4.58·3-s + 0.323i·5-s − 8.09·7-s + 12.0·9-s + 15.7i·11-s − 2.70·13-s + 1.48i·15-s − 23.8·17-s + (−12.2 + 14.5i)19-s − 37.1·21-s − 28.7·23-s + 24.8·25-s + 13.9·27-s − 51.7·29-s − 53.3i·31-s + ⋯ |
| L(s) = 1 | + 1.52·3-s + 0.0647i·5-s − 1.15·7-s + 1.33·9-s + 1.43i·11-s − 0.207·13-s + 0.0989i·15-s − 1.40·17-s + (−0.645 + 0.763i)19-s − 1.76·21-s − 1.24·23-s + 0.995·25-s + 0.515·27-s − 1.78·29-s − 1.72i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9459324038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9459324038\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (12.2 - 14.5i)T \) |
| good | 3 | \( 1 - 4.58T + 9T^{2} \) |
| 5 | \( 1 - 0.323iT - 25T^{2} \) |
| 7 | \( 1 + 8.09T + 49T^{2} \) |
| 11 | \( 1 - 15.7iT - 121T^{2} \) |
| 13 | \( 1 + 2.70T + 169T^{2} \) |
| 17 | \( 1 + 23.8T + 289T^{2} \) |
| 23 | \( 1 + 28.7T + 529T^{2} \) |
| 29 | \( 1 + 51.7T + 841T^{2} \) |
| 31 | \( 1 + 53.3iT - 961T^{2} \) |
| 37 | \( 1 - 30.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 41.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.24iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 1.82T + 2.20e3T^{2} \) |
| 53 | \( 1 - 10.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 13.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 96.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 22.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 88.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 34.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 65.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 120. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 88.2iT - 9.40e3T^{2} \) |
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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
−9.665864687305982588833266415144, −9.203397286494899500683662949729, −8.331963462973757486089109244369, −7.46362050093728646028494306497, −6.85525400902125687939224079902, −5.83818468341983398579724433855, −4.24095811728661570650177076780, −3.85505969170526903180193223398, −2.50407717038734441485194509334, −2.05540065050007918173777463164,
0.19661569567619850170239758534, 2.01578613001298806067337223377, 3.01468784961848399930806898174, 3.54676626210282548041818225165, 4.63430201110303352287937008733, 6.07251984637936369274545503576, 6.73607156010144381840156072892, 7.72243655360295950278512027954, 8.671323818542848695543471125525, 8.957747870719584994902408102332
