L(s) = 1 | + (1.41 + 0.0323i)2-s + (1.99 + 0.0915i)4-s + (1.82 − 1.82i)5-s − 1.81i·7-s + (2.82 + 0.194i)8-s + (2.64 − 2.52i)10-s + (−2.37 + 2.37i)11-s + (−3.20 − 3.20i)13-s + (0.0588 − 2.57i)14-s + (3.98 + 0.365i)16-s − 5.42·17-s + (4.84 + 4.84i)19-s + (3.81 − 3.48i)20-s + (−3.43 + 3.27i)22-s + 4.85i·23-s + ⋯ |
L(s) = 1 | + (0.999 + 0.0228i)2-s + (0.998 + 0.0457i)4-s + (0.817 − 0.817i)5-s − 0.687i·7-s + (0.997 + 0.0686i)8-s + (0.835 − 0.798i)10-s + (−0.715 + 0.715i)11-s + (−0.888 − 0.888i)13-s + (0.0157 − 0.687i)14-s + (0.995 + 0.0914i)16-s − 1.31·17-s + (1.11 + 1.11i)19-s + (0.853 − 0.779i)20-s + (−0.731 + 0.698i)22-s + 1.01i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.66513 - 0.658258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.66513 - 0.658258i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.41 - 0.0323i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.82 + 1.82i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.81iT - 7T^{2} \) |
| 11 | \( 1 + (2.37 - 2.37i)T - 11iT^{2} \) |
| 13 | \( 1 + (3.20 + 3.20i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.42T + 17T^{2} \) |
| 19 | \( 1 + (-4.84 - 4.84i)T + 19iT^{2} \) |
| 23 | \( 1 - 4.85iT - 23T^{2} \) |
| 29 | \( 1 + (0.399 + 0.399i)T + 29iT^{2} \) |
| 31 | \( 1 - 6.64T + 31T^{2} \) |
| 37 | \( 1 + (-3.83 + 3.83i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.62iT - 41T^{2} \) |
| 43 | \( 1 + (4.23 - 4.23i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.76T + 47T^{2} \) |
| 53 | \( 1 + (4.70 - 4.70i)T - 53iT^{2} \) |
| 59 | \( 1 + (-3.25 + 3.25i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.12 + 5.12i)T + 61iT^{2} \) |
| 67 | \( 1 + (1.80 + 1.80i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.84iT - 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 4.10T + 79T^{2} \) |
| 83 | \( 1 + (8.68 + 8.68i)T + 83iT^{2} \) |
| 89 | \( 1 + 5.34iT - 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−11.22629298570794742005544535285, −10.10931828879495538318869829367, −9.659167703787820092129585109894, −8.012557238690046770070585742619, −7.34145889759391873301569811525, −6.09611698848867074169508799718, −5.16074813877386601532392000737, −4.50094459289596980432544235728, −2.98927925505145322721399057636, −1.62652061959464416239744490670,
2.33340092800542584920154987685, 2.83109033878874513921480199062, 4.55291988056571362302810007189, 5.45134584640014033503907175861, 6.48814184086360029568588080054, 7.03434620036917103448656298675, 8.461673653786661224524480684637, 9.622585508915515533516354410107, 10.53387898969201953661802614071, 11.35621127284978395458509183140