| L(s) = 1 | + 2.82i·2-s − 8.00·4-s − 29.6i·5-s − 22.6i·8-s + 83.7·10-s − 12.1i·11-s + 89.7·13-s + 64.0·16-s − 54.0i·17-s + 208.·19-s + 236. i·20-s + 34.4·22-s + 702. i·23-s − 252.·25-s + 253. i·26-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s − 1.18i·5-s − 0.353i·8-s + 0.837·10-s − 0.100i·11-s + 0.531·13-s + 0.250·16-s − 0.187i·17-s + 0.577·19-s + 0.592i·20-s + 0.0712·22-s + 1.32i·23-s − 0.403·25-s + 0.375i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.911821968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911821968\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 29.6iT - 625T^{2} \) |
| 11 | \( 1 + 12.1iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 89.7T + 2.85e4T^{2} \) |
| 17 | \( 1 + 54.0iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 208.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 702. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 746. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 948.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.02e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.86e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.82e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 684. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.85e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 913. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 973.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.43e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.05e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.10e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 6.14e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 44.6iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.24e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.25e3T + 8.85e7T^{2} \) |
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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
−9.479652664661828334740988225702, −8.385046583427015108618092253499, −8.044341756798885490301511999662, −6.89656079980119289585931162619, −5.93497623576297438757216875331, −5.14115867207209387666499814287, −4.37376422487486005281794943135, −3.23896002296848379393057173025, −1.51684737577767157544601791215, −0.53897704332998199855669967242,
0.942046893444407602282520603557, 2.28985745294047364508323925388, 3.10441849870543981868090600688, 4.00518809844396303375580223969, 5.14959887940074352050525776321, 6.30476149984567614456680373025, 6.99918791333554545860940927821, 8.072788179293651634235291507631, 8.905779968196437760770617119496, 9.919065262852120268061626710006
