| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.0603 + 0.342i)3-s + (0.766 + 0.642i)4-s + (1.53 − 1.28i)5-s + (0.0603 − 0.342i)6-s + (0.879 − 1.52i)7-s + (−0.500 − 0.866i)8-s + (2.70 − 0.984i)9-s + (−1.87 + 0.684i)10-s + (−2.11 − 3.66i)11-s + (−0.173 + 0.300i)12-s + (0.184 − 1.04i)13-s + (−1.34 + 1.13i)14-s + (0.532 + 0.446i)15-s + (0.173 + 0.984i)16-s + (−6.69 − 2.43i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.0348 + 0.197i)3-s + (0.383 + 0.321i)4-s + (0.685 − 0.574i)5-s + (0.0246 − 0.139i)6-s + (0.332 − 0.575i)7-s + (−0.176 − 0.306i)8-s + (0.901 − 0.328i)9-s + (−0.594 + 0.216i)10-s + (−0.637 − 1.10i)11-s + (−0.0501 + 0.0868i)12-s + (0.0512 − 0.290i)13-s + (−0.360 + 0.302i)14-s + (0.137 + 0.115i)15-s + (0.0434 + 0.246i)16-s + (−1.62 − 0.591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0534 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0534 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.821635 - 0.866789i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.821635 - 0.866789i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-0.0603 - 0.342i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-1.53 + 1.28i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.879 + 1.52i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.11 + 3.66i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.184 + 1.04i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (6.69 + 2.43i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.87 - 2.41i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (5.98 - 2.17i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.41 + 7.64i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 + (-0.326 - 1.85i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.83 + 2.37i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.45 + 1.25i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (7.57 + 6.35i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-6.85 - 2.49i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-4.06 - 3.41i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 3.98i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.16 + 1.81i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.137 - 0.780i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.16 + 6.59i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.754 - 1.30i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.06 + 11.7i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (1.76 + 0.642i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.11106507086471598394293156498, −9.309965843478943958482356880180, −8.719362090317616046654098527767, −7.67469373564659804630951984161, −6.85337405558913180344681040331, −5.69963337123082062422865135333, −4.68150884345896726489905263866, −3.52375857245775202365666341109, −2.08933183538911546845797259667, −0.76495207097521342370367671049,
1.84752818571102867198932567584, 2.42440842108811029778468413040, 4.37497030464239569246399901584, 5.34309119980350395943933164083, 6.61459809733432029235543431134, 6.98245896754663951371453807090, 8.074815124809273358047682409126, 8.925292750647151125296211019132, 9.810349266891381674220723710470, 10.49359045198842431998087196528
