L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + 3·5-s + (−0.5 + 2.59i)7-s − 0.999·8-s + (1.5 + 2.59i)10-s + 3·11-s + (−2.5 − 4.33i)13-s + (−2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (1.5 + 2.59i)17-s + (−2.5 + 4.33i)19-s + (−1.49 + 2.59i)20-s + (1.5 + 2.59i)22-s + 3·23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (0.474 + 0.821i)10-s + 0.904·11-s + (−0.693 − 1.20i)13-s + (−0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.363 + 0.630i)17-s + (−0.573 + 0.993i)19-s + (−0.335 + 0.580i)20-s + (0.319 + 0.553i)22-s + 0.625·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.52529 + 1.12326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.52529 + 1.12326i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 2.59i)T \) |
good | 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.79266121519492507808887655519, −10.39886497102759487159366782188, −9.619018142985817774990648318467, −8.818895695303352818051336305808, −7.80083606031369702444328819105, −6.41502720249413499461459530719, −5.86709639487708148459070887951, −5.04012728088693649310951558838, −3.38853552349609335559005879725, −2.01824613382574565182518686881,
1.38123911080229985451394679110, 2.69409037696983255698842367186, 4.18234332028656299378009613305, 5.10149594255908444749909301453, 6.47269556162318140505526488840, 7.01373435114625526197951900274, 8.812875209103618643700518907390, 9.647416055390662117375724183757, 10.11373510764729176653577055372, 11.24405396601319947006869578317