| L(s) = 1 | + 3.62·3-s + (0.561 − 4.96i)5-s − 6.45·7-s + 4.12·9-s + 3.94i·11-s + 15.5i·13-s + (2.03 − 17.9i)15-s + 25.4i·17-s + 32.0i·19-s − 23.3·21-s + 15.2·23-s + (−24.3 − 5.57i)25-s − 17.6·27-s − 34.7·29-s − 20.2i·31-s + ⋯ |
| L(s) = 1 | + 1.20·3-s + (0.112 − 0.993i)5-s − 0.921·7-s + 0.458·9-s + 0.358i·11-s + 1.19i·13-s + (0.135 − 1.19i)15-s + 1.49i·17-s + 1.68i·19-s − 1.11·21-s + 0.664·23-s + (−0.974 − 0.223i)25-s − 0.654·27-s − 1.19·29-s − 0.652i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.112 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.887651446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.887651446\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.561 + 4.96i)T \) |
| good | 3 | \( 1 - 3.62T + 9T^{2} \) |
| 7 | \( 1 + 6.45T + 49T^{2} \) |
| 11 | \( 1 - 3.94iT - 121T^{2} \) |
| 13 | \( 1 - 15.5iT - 169T^{2} \) |
| 17 | \( 1 - 25.4iT - 289T^{2} \) |
| 19 | \( 1 - 32.0iT - 361T^{2} \) |
| 23 | \( 1 - 15.2T + 529T^{2} \) |
| 29 | \( 1 + 34.7T + 841T^{2} \) |
| 31 | \( 1 + 20.2iT - 961T^{2} \) |
| 37 | \( 1 + 9.93iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 52.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 21.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 35.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 16.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 77.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 14.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 107. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 93.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 92.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 26.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 27.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 133. iT - 9.40e3T^{2} \) |
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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
−9.452662473646009772125303743459, −8.952131681186541279609131886034, −8.175780278092340978955559724658, −7.48755454956236434071857930440, −6.30202852719503971000236420229, −5.57627705397875139028164988883, −4.04576524343079957061707509128, −3.81123812532256258325653449824, −2.35714269433594286318057546436, −1.49506280525070313040082294926,
0.43961493686993122609258705388, 2.49378801677159716478691997694, 2.95884911714000617913032432588, 3.57013150150281557835181225217, 5.06193621784753543191809887809, 6.08684247615955538907615648023, 7.10857127248432897975795641231, 7.50251107942320144561264275253, 8.590605337924569995968644964973, 9.383442835760191727935361492037
