L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (1.5 − 2.59i)5-s + 0.999·6-s + (2.5 − 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)10-s + (1 + 1.73i)11-s + (0.499 − 0.866i)12-s + 3·13-s + (0.500 − 2.59i)14-s + 3·15-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (0.670 − 1.16i)5-s + 0.408·6-s + (0.944 − 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.474 − 0.821i)10-s + (0.301 + 0.522i)11-s + (0.144 − 0.249i)12-s + 0.832·13-s + (0.133 − 0.694i)14-s + 0.774·15-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.16217 - 1.43823i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16217 - 1.43823i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 + (1.5 + 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4T + 97T^{2} \) |
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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
−9.698978849871421478887554759386, −9.275404821433045568781440017747, −8.420966649392407149401819269377, −7.58650326605607548864565798909, −5.99867157075521024151450942029, −5.32953523068201392874478541384, −4.40271287235392247223877202749, −3.79104943082822696333042728347, −2.11166071264027231088414276182, −1.25888122255994271328450827496,
1.64392197025168130289364477576, 2.85845459494656305809856509170, 3.77775830799584899352373493507, 5.28764471442302278185453342159, 5.90851710386689780904072709609, 6.84000641822528066806791789198, 7.43381893530918799645010840663, 8.438386670447348251852927220398, 9.067577544070458695879792857461, 10.17558176096704126609763165759