L(s) = 1 | + (0.506 − 1.29i)2-s + (−1.61 + 0.622i)3-s + (0.0568 + 0.0527i)4-s + (−3.22 + 1.55i)5-s + (−0.0153 + 2.40i)6-s + (−0.247 − 2.63i)7-s + (2.59 − 1.24i)8-s + (2.22 − 2.01i)9-s + (0.370 + 4.94i)10-s + (2.17 + 2.72i)11-s + (−0.124 − 0.0498i)12-s + (1.48 − 3.77i)13-s + (−3.52 − 1.01i)14-s + (4.24 − 4.51i)15-s + (−0.286 − 3.82i)16-s + (3.20 − 2.97i)17-s + ⋯ |
L(s) = 1 | + (0.358 − 0.912i)2-s + (−0.933 + 0.359i)3-s + (0.0284 + 0.0263i)4-s + (−1.44 + 0.694i)5-s + (−0.00624 + 0.980i)6-s + (−0.0936 − 0.995i)7-s + (0.917 − 0.441i)8-s + (0.741 − 0.670i)9-s + (0.117 + 1.56i)10-s + (0.656 + 0.823i)11-s + (−0.0359 − 0.0143i)12-s + (0.411 − 1.04i)13-s + (−0.942 − 0.271i)14-s + (1.09 − 1.16i)15-s + (−0.0717 − 0.957i)16-s + (0.776 − 0.720i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.832433 - 0.731780i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.832433 - 0.731780i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.61 - 0.622i)T \) |
| 7 | \( 1 + (0.247 + 2.63i)T \) |
good | 2 | \( 1 + (-0.506 + 1.29i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (3.22 - 1.55i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-2.17 - 2.72i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.48 + 3.77i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-3.20 + 2.97i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.138 - 0.240i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.652 + 2.86i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (5.82 + 1.79i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-4.09 + 7.09i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.63 - 1.12i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-7.36 - 5.02i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (-5.99 + 4.08i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (0.963 - 2.45i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (9.47 - 2.92i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-1.71 + 1.16i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (8.47 - 7.86i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-4.49 + 7.78i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.68 - 11.7i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.877 - 0.132i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (4.01 + 6.95i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.83 - 4.67i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (4.14 + 10.5i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-6.74 + 11.6i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.19806347982539105773553893223, −10.39345678277211595178704098340, −9.694303990116702390026373992684, −7.62609355849638910435728046262, −7.44898772650526324534059039532, −6.25678001002284783249666468219, −4.52170764019114834279548821872, −4.00539633931830504319004624098, −3.08128989355332255712998866239, −0.821071685804712059001853879918,
1.40571719732848129259536905397, 3.82064751982817101916948126278, 4.85059055379087366363575155533, 5.78208422180034109376133410028, 6.48622101160868289009416183859, 7.53051844814972411406485899353, 8.279045497306738599728769422650, 9.238430323343197589532005578345, 10.92819015015852158070239628961, 11.41590207924910532722796255127