| L(s) = 1 | − 8.29i·5-s + 8.55i·7-s − 13.7·11-s + 17.0i·13-s − 20.3·17-s + 20.4·19-s + 5.51i·23-s − 43.7·25-s − 41.0i·29-s − 22.2i·31-s + 70.9·35-s + 11.6i·37-s + 35.9·41-s + 66.8·43-s + 19.9i·47-s + ⋯ |
| L(s) = 1 | − 1.65i·5-s + 1.22i·7-s − 1.25·11-s + 1.31i·13-s − 1.19·17-s + 1.07·19-s + 0.239i·23-s − 1.75·25-s − 1.41i·29-s − 0.719i·31-s + 2.02·35-s + 0.314i·37-s + 0.876·41-s + 1.55·43-s + 0.424i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.545812242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545812242\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.29iT - 25T^{2} \) |
| 7 | \( 1 - 8.55iT - 49T^{2} \) |
| 11 | \( 1 + 13.7T + 121T^{2} \) |
| 13 | \( 1 - 17.0iT - 169T^{2} \) |
| 17 | \( 1 + 20.3T + 289T^{2} \) |
| 19 | \( 1 - 20.4T + 361T^{2} \) |
| 23 | \( 1 - 5.51iT - 529T^{2} \) |
| 29 | \( 1 + 41.0iT - 841T^{2} \) |
| 31 | \( 1 + 22.2iT - 961T^{2} \) |
| 37 | \( 1 - 11.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 35.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 66.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 19.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 17.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 62.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 47.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 74.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 16.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 101.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 0.879iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 23.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 16.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 188.T + 9.40e3T^{2} \) |
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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
−8.839373290709430130196936381899, −8.129692923301887103765654079930, −7.38789932215528959878158011179, −6.08603589720549389657186201700, −5.55317444313263031218720117142, −4.74502821625313220561420455014, −4.17263674481552430022142513830, −2.59208060427383449432378426475, −1.90731114799011778675478976787, −0.53905655620381741579346125094,
0.73100815080844850716677567365, 2.41376806077273895844683150620, 3.07492190605509947167884681561, 3.83964343984990162084892389516, 5.00754065160274107361507120450, 5.84513711780194796530989729257, 6.85552799459875124466058041962, 7.35094517349584226674856882620, 7.80537147687981466260126108789, 8.892389510746046191817040715180
