L(s) = 1 | + (−1.05 + 0.610i)2-s + (1.97 − 1.14i)3-s + (−0.253 + 0.439i)4-s + (−1.39 + 2.41i)6-s + 1.28i·7-s − 3.06i·8-s + (1.11 − 1.92i)9-s + 0.285·11-s + 1.15i·12-s + (4.33 + 2.5i)13-s + (−0.785 − 1.35i)14-s + (1.36 + 2.36i)16-s + (5.40 − 3.11i)17-s + 2.71i·18-s + (−2.92 + 3.22i)19-s + ⋯ |
L(s) = 1 | + (−0.748 + 0.431i)2-s + (1.14 − 0.659i)3-s + (−0.126 + 0.219i)4-s + (−0.569 + 0.987i)6-s + 0.485i·7-s − 1.08i·8-s + (0.370 − 0.641i)9-s + 0.0859·11-s + 0.334i·12-s + (1.20 + 0.693i)13-s + (−0.209 − 0.363i)14-s + (0.341 + 0.590i)16-s + (1.30 − 0.756i)17-s + 0.639i·18-s + (−0.671 + 0.740i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.32236 + 0.381355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.32236 + 0.381355i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (2.92 - 3.22i)T \) |
good | 2 | \( 1 + (1.05 - 0.610i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.97 + 1.14i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 - 1.28iT - 7T^{2} \) |
| 11 | \( 1 - 0.285T + 11T^{2} \) |
| 13 | \( 1 + (-4.33 - 2.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.40 + 3.11i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-4.53 - 2.61i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.642 + 1.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + 10.8iT - 37T^{2} \) |
| 41 | \( 1 + (-0.420 - 0.728i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.28 - 2.47i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.96 - 2.86i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (10.7 + 6.18i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.86 - 4.96i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.22 - 3.86i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.853 + 0.492i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.46 - 2.53i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.661 - 0.382i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.72 + 13.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.66iT - 83T^{2} \) |
| 89 | \( 1 + (8.01 - 13.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.1 - 5.87i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.99414959220513975966886082391, −9.678988065313151131798904385688, −9.017864923387313263007061588690, −8.417068975898890075396514785995, −7.65482960035276791627206931287, −6.92161490308140270906688225581, −5.73586482580326957970393264147, −3.96158581462814696809061145997, −2.99644297350544636708875777220, −1.45259554222653824464322057216,
1.20313498701264638439896798879, 2.82617042960674773856305883654, 3.79639384295229104006268568924, 5.02462883313954886470832796372, 6.30676042431863374878835548773, 7.83675431604603771166153610304, 8.537722271054482994534671067193, 9.061296218770222602790075641625, 10.13480595821379948806505530818, 10.48261206172751923644813095200