| L(s) = 1 | + (0.654 + 0.755i)3-s + (−0.321 − 2.23i)5-s + (0.387 − 0.847i)7-s + (−0.142 + 0.989i)9-s + (2.69 + 0.791i)11-s + (−0.948 − 2.07i)13-s + (1.47 − 1.70i)15-s + (0.0340 + 0.0218i)17-s + (7.12 − 4.58i)19-s + (0.894 − 0.262i)21-s + (−4.33 − 2.05i)23-s + (−0.0948 + 0.0278i)25-s + (−0.841 + 0.540i)27-s + (−1.27 − 0.822i)29-s + (4.09 − 4.72i)31-s + ⋯ |
| L(s) = 1 | + (0.378 + 0.436i)3-s + (−0.143 − 0.999i)5-s + (0.146 − 0.320i)7-s + (−0.0474 + 0.329i)9-s + (0.812 + 0.238i)11-s + (−0.263 − 0.576i)13-s + (0.381 − 0.440i)15-s + (0.00825 + 0.00530i)17-s + (1.63 − 1.05i)19-s + (0.195 − 0.0572i)21-s + (−0.903 − 0.428i)23-s + (−0.0189 + 0.00557i)25-s + (−0.161 + 0.104i)27-s + (−0.237 − 0.152i)29-s + (0.735 − 0.848i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59028 - 0.499715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59028 - 0.499715i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (4.33 + 2.05i)T \) |
| good | 5 | \( 1 + (0.321 + 2.23i)T + (-4.79 + 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.387 + 0.847i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-2.69 - 0.791i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.948 + 2.07i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.0340 - 0.0218i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-7.12 + 4.58i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (1.27 + 0.822i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.09 + 4.72i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.406 - 2.82i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.537 + 3.73i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.20 - 8.31i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 1.48T + 47T^{2} \) |
| 53 | \( 1 + (3.20 - 7.02i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.68 + 3.69i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (8.43 - 9.73i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (2.80 - 0.822i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.60 + 0.765i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (4.28 - 2.75i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (1.17 + 2.58i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.773 + 5.38i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (2.23 + 2.57i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.0164 + 0.114i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.60635288137748250278050175376, −9.585706539081832812906571299870, −9.114160077545269869375951354460, −8.076382558587709306591799048952, −7.34402711603559343134313033173, −5.96758111398040706316063581512, −4.83537962737843837980708801085, −4.17993191006694174195845680032, −2.83053274506705573295095464839, −1.06114576218818220774911134006,
1.64832034829251260780909626289, 3.00825017493484071142703355542, 3.89521647058148058575331974574, 5.47475214620639329922210548627, 6.49628623814963617124385233982, 7.25704479196985363733231173764, 8.080757267454458663149523824204, 9.151973232459992552667484591611, 9.908239907449940520342182032883, 10.94573885345124013501741805152
