L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−1.78 + 1.02i)5-s + (0.289 + 2.62i)7-s + 0.999·8-s + (1.78 + 1.02i)10-s + (1.29 + 3.05i)11-s − 6.97i·13-s + (2.13 − 1.56i)14-s + (−0.5 − 0.866i)16-s + (−2.66 + 4.61i)17-s + (7.36 − 4.25i)19-s − 2.05i·20-s + (1.99 − 2.64i)22-s + (−1.35 + 0.782i)23-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.797 + 0.460i)5-s + (0.109 + 0.993i)7-s + 0.353·8-s + (0.563 + 0.325i)10-s + (0.390 + 0.920i)11-s − 1.93i·13-s + (0.569 − 0.418i)14-s + (−0.125 − 0.216i)16-s + (−0.646 + 1.11i)17-s + (1.69 − 0.975i)19-s − 0.460i·20-s + (0.425 − 0.564i)22-s + (−0.282 + 0.163i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6358795645\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6358795645\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.289 - 2.62i)T \) |
| 11 | \( 1 + (-1.29 - 3.05i)T \) |
good | 5 | \( 1 + (1.78 - 1.02i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 6.97iT - 13T^{2} \) |
| 17 | \( 1 + (2.66 - 4.61i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-7.36 + 4.25i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.35 - 0.782i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.87T + 29T^{2} \) |
| 31 | \( 1 + (4.39 - 7.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.82 + 3.15i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 2.42T + 41T^{2} \) |
| 43 | \( 1 - 7.55iT - 43T^{2} \) |
| 47 | \( 1 + (10.2 - 5.91i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (9.93 + 5.73i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.49 - 2.01i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.39 - 2.53i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.75 - 8.23i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.30iT - 71T^{2} \) |
| 73 | \( 1 + (0.828 + 0.478i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.75 + 4.47i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 + (-2.99 + 1.72i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.95T + 97T^{2} \) |
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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
−9.838269798249630487276394529958, −9.094337196152137048691312025536, −8.192804644815906517613774376695, −7.65006002688825642401521735091, −6.72973389427531600289724734814, −5.53142164760143991884291501104, −4.71895594739827292672564744357, −3.41637906156816638533760735898, −2.88538967555270951433430996636, −1.50630574278993880721126828877,
0.30912685784572215564667972847, 1.59324272918527313543605207077, 3.51060627612020860381084585029, 4.26106757730394291278959841287, 5.04263315167911549502420947791, 6.27648045965203895940735952601, 6.99797386215829484605497056307, 7.70134123842224370653903092899, 8.413824985894751791988717088352, 9.301729175083457526291096738868