L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.563 + 0.975i)3-s + (−0.499 − 0.866i)4-s + (−0.858 + 1.48i)5-s − 1.12·6-s + (0.779 + 2.52i)7-s + 0.999·8-s + (0.865 − 1.49i)9-s + (−0.858 − 1.48i)10-s + (1.82 + 3.16i)11-s + (0.563 − 0.975i)12-s − 3.79·13-s + (−2.57 − 0.589i)14-s − 1.93·15-s + (−0.5 + 0.866i)16-s + (−2.08 − 3.61i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.325 + 0.563i)3-s + (−0.249 − 0.433i)4-s + (−0.383 + 0.664i)5-s − 0.459·6-s + (0.294 + 0.955i)7-s + 0.353·8-s + (0.288 − 0.499i)9-s + (−0.271 − 0.469i)10-s + (0.550 + 0.953i)11-s + (0.162 − 0.281i)12-s − 1.05·13-s + (−0.689 − 0.157i)14-s − 0.499·15-s + (−0.125 + 0.216i)16-s + (−0.506 − 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.352959 + 0.990320i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.352959 + 0.990320i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-0.779 - 2.52i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.563 - 0.975i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.858 - 1.48i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.82 - 3.16i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.79T + 13T^{2} \) |
| 17 | \( 1 + (2.08 + 3.61i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.06 - 7.03i)T + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 + (4.27 + 7.39i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.46 - 7.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 8.11T + 43T^{2} \) |
| 47 | \( 1 + (-1.47 + 2.55i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.52 + 2.64i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.09 + 3.62i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.65 + 9.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.61 - 11.4i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 + (-2.13 - 3.69i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.43T + 83T^{2} \) |
| 89 | \( 1 + (1.97 - 3.42i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.121T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.01969256736618525247335337242, −10.89794731296932183924841764678, −9.747464684196289197873414143774, −9.353084663115143132244692856671, −8.207834751990389450482045437475, −7.21210868980448516427448219981, −6.34668248360656769033519828969, −4.97467168129973175537505836739, −3.92733476383879010713317471781, −2.30989076152133357370854176806,
0.848211961855845004252168154450, 2.35886412154525801782841577629, 4.00441012442163824843701266131, 4.87019046319939083824560049358, 6.74981756325710581704374945500, 7.60042107934937517874990301275, 8.513650633181264891870352955240, 9.189745522491717400858744414835, 10.73535574711261339982988691720, 10.93710108247038157706517981268